T J 

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Class _Til2.M 
CopghtN" 



COPYRIGHT DEPOSm 



J 



NOTE S 



ON 



MACHINE DESIGM 



BY 

• / 

CHARLES H. BENJAMIN, M. E, 



PROFESSOR OF MECHANICAL ENGINEERING, 



CASE SCHOOL. OF APPLIED SCIENCE. 



}^-. I 



SECOND EDITION. 



* • • • < 
• • • • • 



COPYRIGHTED 



• • • 






• ' • -» • 



CLEVELAND : 
ChARI^ES H. HOI^MES, PUBI.ISHER, 2303 EUCI.ID AvBw 

1902. 



T3" zi o 



THE LIBRARY OF 

CONGRESS, 
Oni Oopy Reccivco 

FEB. 24 1902 

CofrrmoHT rwtry 

C1LAS$ ^ XXc No. 

OOPY B. 




m « 
• « « 




©xjrnt^nt^* 



OHARTER 7. 

Page. 

Units used. Materials, properties and strength. Nota- 
tion. Formulas. Constants of cross-sections. Formulas for 
loaded beams 3 

CHAPTER 2, 

General principles governing the design of frames and 
supports 12 

CHAPTER 3. 

Stationary Machine Members: 

Thin and thick shells. Steam, gas and water pipe. 
Cast-iron steam cylinders. Flat plates. Machine frames. i6 

CHAPTER 4. 

Springs : 

Tension and compression. Torsion. Flat or leaf 
springs 31 

CHAPTER a. 

Fastenings: 

Bolts and nuts. Riveted joints. Joint pins and cotters. 39 

CHAPTER e, 

S1.1DING Bearings: 

General rules. Angular slides. Gibbed slides. Flat 
slides. Circular guides. Stuffing-boxes .... . r^*-, . 53 

CHAPTER 7. 

jouRNAi^, Pivots and Bearings : 

Adjustment, Lubrication. Friction. I/imits of pressure. 
Heating. Strength and stiffness. Caps and bolts. Friction 
of pivots. Conical pivot. Schiele's pivot. Collar bearings. 61 



CHARTER 8. 

Bai.1. and R01.1.ER Bearings: 

General principles. Journal and step ball bearings. 
Materials and wear. Design. Roller bearings. Hyatt 
rollers. Roller steps 76 

CHARTER a. 

Shafting, Couplings and Hangers : 

Strength of shafting. Couplings. Coupling bolts and 
keys. Hangers and boxes 83 

CHARTER JO. 

Gears, Pui.i.evs and Fi.y-WheeivS. 

Gear teeth. Proportions and strength. Experimental 
data. Teeth of bevel gears. Rim and arms. Safe speed for 
wheels. Bursting of fly-wheels. Rims of gears 91 

CHARTER n. 

Transmission by Bei.ts and Ropes. 

Friction of belting. Strength of belting. Rules for 
horse -power. Centrifugal tension. Manila rope trans- 
mission. Rules and tables. Wire rope transmission. . . 112 



preface to ^jeconif @Mtf xtn. 



In presenting this book no claim is made of 
^^, originality of subject matter, as nearly every 
^ thing in it can be found elsewhere. The object 
in preparing the book was to gather together in small 
compass the more simple formulas for the strength and 
stiffness of machine parts, with an explanation of the 
principles involved, and with such tables and general 
information as the designer of machinery might find 
useful. 

The book pre-supposes an acquaintance with math- 
ematics and the laws of the strength of materials. 

In short, the aim has been to put the mathematical 
principles of macnine design in a compact form at a 
moderate price for the use of the student and the young 
engineer. 

In revising the text for a second edition some 
additions have been made to the physical constants in 
tables I and II as the result of recent experiments. 
Experimental data obtained by the author in the labor- 
atories of the school have also been added, notably 
those in regard to iron and steel pulleys, belts, fly 
wheels, gear teeth, ball bearings, and the friction of 
steam packings. 

The author wishes to acknowledge the great assist- 
ance given him by Mr. J. Verne Stanford in the 
preparation of drawings for the cuts in this edition. 



(S^haptev I* 



UNITS AND TABLES. 

1. Units. In this book the following units will 
be used unless otherwise stated. 

Dimensions in inches. 

Forces in pounds. 

Stresses in pounds per square inch. 

Velocities in feet per second. 

Work and energy in foot pounds. 

Moments in pounds inches. 

Speeds of rotation in revolutions per minute. 

The word stress will be used to denote the resist- 
ance of material to distortion per unit of sectional area. 
The word strain will be used to denote the distortion 
of a piece per unit of length. The word set will be 
used to denote total permament distortion of a piece. 

In making calculations the use of the slide-rule and 
of four-place logarithms is recommended ; accuracy is 
expected only to three significant figures. 

2. Materials. The principal materials used in 
machine construction are given in the following tables 
with the physical characteristics of each. 

By wrougt iron is meant commercially pure Iron 
which has been made from molten pig-iron by the 
puddling process and then squeezed and rolled, thus 
developing the fiber. This iron has been largely sup- 
planted by soft steel. 

In making steel, on the other hand, the molten 
iron has had the silicon and carbon removed by a hot 
blast, either passing through the liquid as in the Bes- 
semer converter, or over its surface as in the open-hearth 
furnace. A suitable quantity of carbon and manganese 
has then been added and the metal poured into ingot 
molds. If the steel is then reheated and passed throu g h 



4 MACHINE DESIGNo 

a series of rolls, structural steel and rods or rails result. 
Steel castings are poured directly from the open 
iiearth furnace and allowed to cool without any draw- 
ing or rolling. They are coarser and more crystalline 
than the rolled steel. 

Open hearth steel is generally used for boiler plates 
and of these, two grades are commonly known as 
marine steel and flange steel. 

Bessemer steel is largely used in the manufacture 
of rails for steam and electric roads. 

Crucible steel usually contains from one to one 
and a half per cent of carbon, is relatively high priced 
and only used for cutting tools. It is made by melting 
steel in an air tight crucible with the proper additions 
of carbon and manganese. 

Cast iron is ma^e directly from the pig by remelt- 
ing and casting, is granular in texture and contains 
from two to five per cent, of carbon. A portion of the 
carbon is chemically combined with the iron while the 
remainder exists in the form of graphite. The harder 
and whiter the iron the more carbon is found chemi- 
cally combined. Silicon is an important element in 
cast iron and influences the rate of cooling. The more 
slowly iron cools after melting the more graphite forms 
and the softer the iron. 

Malleable iron is cast iron annealed and partially de- 
carbonized by being heated in an annealing oven in con- 
tact with some oxidising material such as haematite ore. 
This process makes the iron tougher and less brittle 

All castings including those made from alloys are 
somewhat unreliable on account of hidden flaws and 
of the strains developed by shrinkage while cooling. 

The constants for strength and elasticity are only 
fair average values, and should be determined for an^ 
special material by direct experiment when it is prac- 
ticable. Many of the constants are not given in the 
table on account of the lack of reliable data for their 
determination. 



MACHINE DESIGN. 



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MACHINE DESIGN. 





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MACHINE DESIGN. 7 

3. Notation. 

Let S = Stress per square inch, 

W== Total load applied in pounds. 
M = Bending moment in pounds inches. 
T = Twisting moment in pounds inches. 
b = Breadth of cross -section in inches. 
h=: Depth of cross-section in inches. 
d = Diameter of circular section in inches. 
A=Area of cross-section in square inches. 
l=Ivength of piece in inches. 
I = Rectangular moment of inertia. 
J = Polar moment of inertia. 
c = Half depth of beam or shaft in inches. 
r= Radius of gyration of section in inches. 

— = Section modulus for bending. 

—= Section modulus for twisting. 

4. Formulas. 

Simple Stress. 

Tension, compression or shear, S=-t- (i) 

Be7iding under Transverse Load, 

SI 
General equation, M= — (2) 

Rectangular section, M= — - — (3) 

Rectangular section, bh^= — - — (4) 

Sd^ 

Circular section, M= (s) 

' 10^2 ^^^ 

Circular section, ' d=3 r^-^^ (5) 

Torsion or Twisting, 

S T 
General equation, T=— ^ (7) 



8 



MACHINE DESIGN. 



Sd^ 



5-1 



-n 



S.I 



s 
d^-d,* 



..(8) 

.(9) 
.(id) 



Circular section, T 

Circular section, d 

Hollow circular section, 

Other values of — and — may be taken from Table 4. 
c c 

Combined Bending and Twisting, 

Calculate shaft for a twisting moment, 

T^=M + n/M' + T' (11) 



Column subject to Bending, 
W 



Use Rankine's formula, -^= 



i+q 



r 



(12) 



The values of r^ may be taken from Table IV. The 
subjoined table gives the average values of q, while 
S is the compressive strength of the material. 



TABLE III,— Values of q in formula 12. 


Material. 


Both 
ends 
fixed. 


Fixed 

and 

round. 


Both 
ends 
round. 


Fixed 
and 

free. 


Timber 

Cast Iron 

Wrought Iron... 
Steel 


I 


1.78 
3000 

1.78 
5000 

1.78 
36000 

1.78 

25000 


4 


16 
3000 

t6 
5000 

16 
36000 

16 
25000 


3000 

I 


3000 
4 


5000 


5000 

4 
36000 

4 
25000 


36000 

I 


25000 



MACHINE DESIGN. 9 

In this formula, as in all such, the values of the 

constants should be determined for the material used 

by direct experiment if possible. 

W 1 

Or use straight line formula, -r- =S-— k — (12a) 

A r 



TABLE Ilia. — Values of S and k in fornnula (12a). 

(Memman's Mechanics of Materials.) 


Kind of Column. 


S 


k 


Limit — 
r 


Wrought Iron : 

Flat ends 


42000 
42000 
42000 

52500 
52500 
52500 

80000 
80000 
80000 

5400 


128 

157 
203 

179 

220 
284 

438 
537 
693 

28 


218 
178 
138 

195 
159 
123 

122 
99 

77 

128 


Hinp'ed ends 


Round ends 


Mild Steel : 

Flat ends 


Hinged ends 

Round ends 


Cast Iron : 

Flat ends 

Hinged ends , 

Round ends 


Oah : 

Flat ends 





See also Carnegie's Pocket Companion (pp. 129, 
147 and 152) for applications of these formulas. 

For values of — less than 90 mild steel columns 

are calculated for direct compression. 



lO 



MACHINE DESIGN. 



TABLE 


IV.— CONSTANTS 


OF CROSS-SECTION. 


Form of 


Square of 


Moment 


Section 


Polar 


Tortion 


Section 


Radius of 


of 


Mod'lus 


Moment 


ModUm 


and Area 


Gyration. 


Inertia 


I 


of Iner- 


J 


A 


r' 


I=Ar^ 


c 


tia. J 


c 


Rect'ngle 
bh 


12 


bh2 

12 


bh2 

6 


bh»H- bh3 


bh3 + b^h 


12 


^v/ba2th2 


Square 
d^ 


d2 
12 


d* 

12 


d3 

6 


d^ 

6 


d3 
4.24 


Hollow 












Rect'ngle 
or I-beam 


bh3-b,h? 


bh3-bihf 


bh3-blhf 






(12bh-bihi) 


12 


6h 


bh— b,h, 












Circle 


d2 


Trd^ 


d3 


7rd4 


d3 


'd. 


16 


64 


10.2 


32 


5.1 


4 ' 












Hollow 












Circle 
-^(d.-dD 


d2+ df 
i6 


7r(d^-dt) 


d^-df 
10.2d 


7r(d4-df) 


d^-df 
5.1 d 


64 


32 


Ellipse 

%b 

4 


a2 
16 


7rba3 
64 


ba2 
10.2 


TTba' + ab^ 


ba3- ab3 


64 


10.2 a 



Values of I and J for more complicated sections 
can be worked out from those in table, 



MACHINE DESIGN. 



I*: 



TABLE V.-FORMULAS FOR LOADED BEAMS. 


Beams of Uniform Cross-section. 


Maxi- 
mum 
Moment 

M 


Maxi- 
mum 

Deflec- 
tion 

A 


Cantilever, load at end 

Cantilever uniform load 


Wl 
Wl 

2 
Wl 

4 

Wl 

8 

3W1 
i6 

Wl 
8 

Wl 
8 

Wl 

12 

Wl 
2 


WV 
3E1 

Wl' 

8EI 

WV 

48Ei 

5WI' 
384 E I 

.0182 WP 
Ei 

.0054Wr 


Simple beam, load at middle 

Simple beam, uniform load 

Beam fixed at one end, supported 
at other, load at middle 


Beam fixed at one end, supported 
at other, uniform load 


Ei 

WV 
192 Ei 

WV 
384 Ei 

WV 

"l"2El 


Beam fixed at both ends, load at 
middle , 


Beam fixed at both ends, uniform 
load 

Beam fixed at both ends, load at 
one end, (pulley arm) 



The maximum deflection of cantilevers and beams 
of uniform strength is greater than when the cross- 
section is uniform, fifty per cent, greater if the breadth 
varies, and one hundred per cent greater if the depth 
varies. 



FRAME DESIGNS. 

5. General Principles. The working or moving- 
parts should be designed first and the frame adapted 
to them. 

The moving parts can be first arranged to give 
the motions and velocities desired, special attention 
being paid to compactness and to the convenience of 
the operator. 

Novel and complicated mechanisms should be 
avoided and the more simple and well tried devices 
used. 

Any device which is new should be first tried in a 
working model before being introduced in the design. 

The dimensions of the working parts for strength 
and stiffness must next be determined and the design 
for the frame completed. This may involve some 
modification of the moving parts. 

In designing any part of the machine, the metal 
must be put in the line of stress and bending avoided 
as far as possible. 

Straight lines should be used for the outlines of 
pieces exposed to tension or compression, circular cross 
sections for all parts in tortion, and curves of uniform 
stress for pieces subjected to bending action. 

Superfluous metal must be avoided and this ex- 
cludes all ornamentation as such. There should be a 
good practical reason for every pound of metal in the 
machine. It may be sometimes necessary to waste 
metal in order to save labor in finishing, and in general 
the aim should be to save labor at the expense of the 
stock. 

It is thus necessary for the designer to be familiar 
with all the shop processes as well as the principles of 



MACHINE DESIGN. 1 3 

strength and stability. The usual tendency in design, 
especially of cast iron work, is towards unnecessary 
weight. 

All corners should be rounded for the comfort and 
convenience of the operator, no cracks or sharp inter- 
nal angles left where dirt and grease may accumulate, 
and in general special attention should be paid to so 
designing the machine that it may be safely and con- 
veniently operated, that it may be easily kept clean, and 
that oil holes are readily accessible. The appearance of 
a machine in use is a key to its working condition. 

Polished metal should be avoided on account of its 
tendency to rust, and neither varnish nor bright colors 
tolerated. The paint should be of some neutral tint 
and have a dead finish so as not to show scratches or 
dirt. 

Beauty is an element of machine design, but it can 
only be attained by legitimate means which are appro- 
priate to the material and the surroundings. 

Beauty is a natural result of correct mechanical 
construction but should never be made the object of 
design. 

Harmony of design may be secured by adopting 
one type of cross-section and adhering to it through- 
out, never combining cored or box sections with ribbed 
sections. In cast pieces the thickness of metal should 
be uniform to avoid cooling strains, and for the same 
reason sharp corners should be absent. When aper- 
tures are cut in a frame either for core-prints or for 
lightness, the hole or aperture should be the symetrical 
figure, and not the metal that surrounds it, to make 
the design pleasing to the eye. 

Machine design has been a process of evolution. 
The earher types of machines were built before the 
general introduction of cast iron frames and had frames 
made of wood or stone, paneled, carved and decorated 
as in cabinet or architectural designs. 

When cast iron frames and supports were first 
introduced they were made to imitate wood and stone 



14 MACHINE DESIGN. 

construction, so that in the earher forms we find panels, 
moldings, gothic traceries and elaborate decorations of 
vines, fruit and flowers, the whole covered with con- 
trasting colors of paint and varnished as carefully as a 
piece of furniture for the drawing-room. Relics of this 
transition period in machine architecture may be seen 
in almost every shop. One man has gone down to pos- 
terity as actually advertising an upright drill designed 
in pure Tuscan. 

6. Machine Supports. The fewer the number of 
supports the better. Heavy frames, as of large engines, 
lathes, planers, etc., are best made so as to rest directly 
on a masonry foundation. Short frames as those of 
shapers, screw machines and milling machines, should 
have one support of the cabinet form. The use of a 
cabinet at one end and legs at the other is offensive to 
the eye being inharmonious. If two cabinets are used 
provision should be made for a cradle or pivot at one 
end to prevent twisting of the frame by an uneven 
foimdation. The use of intermediate supports is always 
to be condemned, as it tends to make the frame conform 
to the inequalities of the floor or foundation on what 
has been aptly termed the ' 'caterpillar principle' ' . 

A distinction must be made between cabinets or 
supports which are broad at the base and intended to be 
fastened to the foundation, and legs similar to those of 
a table or chair. The latter are intended to simply rest 
on the floor, should be firmly fastened to the machine 
and should be larger at the upper end where the great- 
est bending moment will come. 

The use of legs instead of cabinets is an assumption 
that the frame is stiff enough to withstand all stresses 
that come upon it, unaided by the foundation, and if 
that is the case intermediate supports are unnecessary. 

Whether legs or cabinets are best adapted to a cer- 
tain machine the designer must determine for himself. 

Where two supports or pairs of legs are necessary 
under a frame, it is best to have them set a certain 
distance from the ends, and make the overhanging part 



MACHINE DESIGN. 1 5 

of the frame of a parabolic form, as this divides up the 
bending moment and allows less deflection at the center. 
Trussing a long cast-iron frame with iron or steel rods 
is objectionable on account of the difference in expan- 
sion of the two metals and the liabihty of the tension 
nuts being tampered with by workmen. 

The sprawling double curved leg which originated 
in the time of Louis XIV and which has served in turn 
for chairs, pianos, stoves and finally for engine lathes 
is wrong both from a practical and aesthetic standpoint. 
It is incorrect in principle and is therefore ugly. 

Exercise i. — Apply the foregoing principles in 
making a written criticism of some engine or machine 
frame and its supports. 



(S^lfaptev 3* 



STATIONARY MACHINE MEMBERS, 



Thin Shells 




Fig. I 



Let Fig. I represent a section of a 
thin shell, like a boiler 
shell, exposed to an inter- 
nal pressxH-e of p pounds 
per' sq. inch. Then, if 
we consider any diameter 
B AB, will the total upward 
pressure on upper half of 
the shell balance the total 
downward pressure on 
the lower half and tend 
to separate the shell at A 
and B by tension. 



I^t d= diameter of shell in inclies. 
r=radius of shell in inches. 
1= length of shell in inches. 
t= thickness of shell in inches. 
S= tensile strength of material. 

Draw the radial line CP to represent the pressure 
on the element P of the surface. 

Area of element at P=lrd(? 
Total pressure on element =plrd<^. 
Vertical pressm-e on element =plr sin 6d0. 

Total vertical pressure on APB=^ I plr sin 6d6--—2plT 

The area to resist tension at A and B=2tl and its 
total strength = 2 tlS. 

Equating the pressure and the resistance 
2tlS = 2plr 
pr_pd 



t = 



2S 



(13) 



MACHINE DESIGN. 1 7 

The total pressure on the end of cyUnder==:rr^p 
and the resistance of a circular ring of metal to this 
pressiu:e=27rrtS 

2:rrSt=:7rr'p 

t- PL-Pi fTA^ 

Therefore a shell is twice as strong in this direc- 
tion as in the other. Notice that this same formula 
would apply to spherical shells. 

In calculating the pressure due to a head of water 
equals h, the following formula is useful: 

p=o.434h (15) 

exampi.es. 

1 . A cast-iron water pipe is 1 2 inches in diameter 
and the metal is .45 inches thick. What would be the 
factor of safety, with an internal pressure due to a head 
of water of 250 feet? 

2. What would be the stress caused by bending 
due to weight, if the pipe in Ex. i were full of water 
and 24 feet long, the ends being merely supported? 

3. A standard lap- welded steam pipe, 8 inches in 
nominal diameter is 0.32 inches thick and is tested with 
an internal presstue of 500 pounds per sq. inch. What 
is the bursting pressure and what is the factor of safety 
above the test pressure, assuming 8=40000? 

7. Thick Shells. There are several formulas for 
thick cylinders and no one of them is entirely satis- 
factory. It is however generally admitted that the 
tensile stress in such a cyhnder caused by internal pres- 
sure is greatest at the inner circumference and dimin- 
ishes according to some law from there to the exterior 
of the shell. This law of variation is expressed differ- 
ently in the different formulas. 

Barlow's Formulas. Here the cylinder diameters 
are assumed to increase under the pressure, but in such 
a way that the volume of metal remains constant. 
Experiment has proved that in extreme cases this last 
assumption is incorrect. Within the limits of ordinary 



18 



MACHINE DESIGN. 



TABLE VI.— WROUGHT IRON WELDED TUBES, 

For Steam, Gas or Water. 



y^ to 14 inclusive, butt-welded, tested to 300 lbs. 
per sq. inch hyraulic pressure. 

I ^/i inch and upwards, lap-welded, tested to 500 
lbs. per sq. inch hydraulic pressure. 



o 
S 

n' 

0) 



W- O 



p 



Pa 



0) 




g' 


^ 


P 


tf> 





(V 


P- 
P 








P- 

0) 






?- 


2 
p 

s 


^ 


0) 

•-t 




* 


w 





o 



W 
n 



O) 
P 
p. 

w 






w 

P-* 

(T) 
P 



74. 
3/8 

1/ 



I 

2 

2^ 

3 

3'A 

4 

4^ 

4 
6 

7 
8 

9 
10 



.40 


•27 


.24 


27 


■54 


•36 


•42 


18 


.67 


•49 


•56 


18 


.84 


.62 


•85 


14 


1,05 


.82 


1. 12 


14 


I-3I 


I 04 


1.67 


11^ 


1.66 


138 


2.25 


11% 


1.90 


1. 61 


2.69 


11^ 


2-37 


2 06 


3.66 


ii'A 


2.87 


2,46 


577 


8 


3-50 


3.06 


7^54 


8 


400 


3-54 


9-05 


8 


450 


4.02 


10 72 


8 


5.00 


450 


12.49 


8 


5-56 


504 


1456 


8 


6.62 


606 


18.77 


8 


7.62 


7.02 


2341 


8 


8.62 


7.98 


28.35 


8 


9.68 


9.00 


3407 


8 


10.75 


10.01 


40.64 


8 



.0572 
.1018 
.1886 
.3019 
.5281 

.8495 
1.4956 

2.0358 

33329 
47329 
7-3529 
9.8423 

12.6924 

15-9043 
199504 

28.8426 

387048 

50.0146 
63.6174 

80.1186 



MACHINE DESIGN. I9 

practice it is, however, approximately true. 

Let d^ and d^ be the interior and exterior diam- 
eters in inches and let t=-^ be the thickness. 

2 

lyCt 1 be the length of cylinder in inches. 
Let Si and Sg be the stresses in lbs. per sq. inch 
at inner and outer circumferences. 
Then it may be proved that 

or the stresses vary inversely as the squares of the cor- 
responding diameters. 

Integrating, the total stress on the area 2tl is found to be 

^='^^'^i (^) 

Equating this to the pressure which tends to produce 

rupture, pdl, where p is the internal unit pressure, 

2S t 

there results : * P=^i — r — (16) 

di+ 2t ^ 

Lame"s Formula, — In this discussion each particle 
of the metal is supposed to be subjected to radial com- 
pression and to tangential and longitudinal tension and 
to be in equilibrium under these stresses. 

Using the same notation 
as in previous formula : 




d/-dx^ 



Pi 



(17) 



Fig. 2. 



for the maximum stress 
Qat the interior. 

andS,=-pf-f^p,..(i8) 

for the stress at the outer 
surface, f 

Fig. 2 illustrates the 
variation in S from inner 
to outer surface. 



For discussion see Merriman's Mechanics of Materials: * p. 26; t pp. 310-14. 



20 MACHINE DESIGN. 

Solving for d, in (i8) we have 

^=-■^IPI ^-> 

exampi.es. 

1. A hydraulic cylinder has an inner diameter of 
8 inches, a thickness of four inches and an internal 
pressure of 1500 lbs. per sq. in. Determine the maxi- 
mum stress on the metal by Barlov^'s and Lame"s 
formulas. 

2. Design a cast iron cylinder 6 inches internal 
diameter to carry a working pressure of 1 200 lbs. per 
sq. in. with a factor of safety of 10. 

3. A cast iron water pipe is i inch thick and 12 
inches internal diameter. Required head of water 
which it will carry with a factor of safety of 6. 

8. Steam Cylinders. Cylinders of steam engines 
can hardly be considered as coming under either of the 
preceeding heads. On the one hand the thickness of 
metal is not enough to insure rigidity as in hydraulic 
cylinders, and on the other the nature of the metal 
used, cast iron, is not such as to warrant the assumption 
of flexibility, as in a thin shell. Most of the formulas 
used for this class of cylinder are empirical and founded 
on modern practice. 

Va7i Bureri's formula for steam cylinders is : 

^ .0001 pd + .i5^/~d~ (20) 

A formula which the writer has developed is 
somewhat similar to Van Buren's. 

lyct s'=tangential stress due to internal pressure. 

Then by equation for thin shells 

2t 

Let s" be an additional tensne stress due to distor- 
tion of the circular section at any weak point. 

Then if we regard one-half of the circular section 
as a beam fixed at A and B (Fig. 3) and assume the 

* See Whitham's "Steam Engine Design", p. 27. 



MACHINE DESIGN. 



21 



maximum bending moment as at C some weak point, 

the tensile stress on the 

outer fibres at C due to 

the bending will be pro- 

pd^ 
portional to-^^^ by the 

Q laws of flexure, or 
cpd^ 




s"= 



un- 



where c is some 
known constant. 

The total tensile stress 
at C will then be 
, cpd^ 



Solving for c c=— r^ 



(a) 



Solving for t 



cpd' pM^ 
S i6S' 



(21) 



a form which reduces to that of equatiop (13) when c=o. 

An examination of several engine cylinders of 
standard manufacture shows values of c ranging from 
.03 to .10, with an average value : 

c = .o6 

The formula proposed by Professor Barr, in his 
recent paper on * ''Current Practice in Engine Propor- 
tions'', as representing the average practice among 
builders of low speed engines is : 

t=.05 d-+-.3 inch ,.,...(22) 

Experiments made at the Case School of Applied 
Science in 1896-97 throw some light on this subject. 
Cast iron cylinders similar to those used on engines 
were tested to failure by water pressure. The cylinders 
varied in diameter from six to twelve inches and in 
thickness from one-half to three-quarters inches. 

Contrary to expectations most of the cylinders 

* Transactions. A. S. M. E,, vol. xviii, p. 741. 



22 



MACHINE DESIGN. 



failed by tearing around a circumference just inside 
the flange. 

In Table VII are assembled the results of the 
various experiments for comparison. The values of S 
by formula (13) are calculated for each cylinder, and by 
formula (14) for all those which failed on a circum- 
ference. It will be noticed that six out of nine faiied 
in the latter way. 





t+H 



























m t/5 


'v/i 


'Ji tn 


t/3 


(/3 


7) 


C/3 






r^ U 


Xi ^ 


^ 


rd ^ 


^ 


rO 


rQ 


^ 






f oJ 


^-H T- 


^ 


'"' 


»— 1 y- 


^ 


•"^ 


'""' 


'■"' 


'-"' 






bJO^ 





Q 


8 8 




















8 8 


g 


g 


g 


g 


g 






00 ^ 


^ 


^ ^ 


^ 


^ 


'^ 


^ 






CO^ 


M CN 


c< 


(N (N 


<N 


<N 


(N 


CN 






oj II 


VsO t^ 


00 


10 o\ 


10 


01 


00 


00 






^ -^ 


^ 


10 ^ 


LO 


t^ 


^ 


CN 






o 


q 


q 


q q 


q 


q 


q 


q 






C/3 
























'O ^ 
























- II 
































t/1 


t^ 




10 










''^t- 






00 






Cj 


^ 




On 


On 




CO 


t^ 






1— ( 






r— ^ 

3 


tn 


ro 




-^ 


UO 




-^ 


CO 






10 


LO 




s 

























'O +^ 




















• 





Cu ^ 


C 


^ 





c 


:> 














11 


__ 


^ w 


'^ 


3 





L 





On 


t^ 


00 


10 


> 


OS t 


~^ 


0^ 


00 H 


■H 


10 


^ 


VO 


SP 


(J 






m 


VO t 


■^ 


C^ 


M t 


■^ 


00 


t^ 


t^ 







UJ 










1— ( 










M 


<-t-l 

























CO 


^2i 


S ., 


■H 


S 


a ., 


_; 


3 


a 


• tH 


a" 


(U 


<r 


ine 
ailu 


a \ 


5J0 


T^ 


3 t 


ji) 


^ 


^u 


biO 


^ 


bX) 


h- 


.^ ^ 


3 
D 


g 


^ \ 


-1 
3 


u 

•T-t 


a 

• 1— t 


a 









h4P^ 


h 


:r 





K 


:j 








.-r 


a 


> 

< 


, 






















-tI ^ 








c 


^j 


CO 


VO 


M 


M 






.Si ^ ^ 


vi 


D 


M 


10 ( 


D 


r^ 


On 


t>» 


CO 








!>- I 





^ 


VO . 


^ 


LO 


10 


10 


10 


























Pres- 
sure. 

P 


( 


3 


10 


( 


D 





LO 


(3 


LO 




( 


3 


CN 


( 


D 


UO 


r^ 





t^ 






00 i 


^>» 


CO 


to vj 


D 





ON 


t^ 


00 










H-1 


cs 




M 






















10 










B 


vO 1 


LO 


M 


<N C 





t^ 


CO 


CO 


VO 






rt nj 


1— 1 


^ 


l-H 


HH \ 





CO 


h-l 


10 


LO 






S 


O) ( 


N 


Cn 


VO ( 


:> 


On 


On 


(N 


oi 






t— t 


— 1 












M 


M 




6 


cJ -1 


d 


CD 


«-^-( 


- 


M 


CO 


^ 


•o 



MACHINE DESIGN. 23 

This appears to be due to two causes. In the first 
place, the influence of the flanges extended to the 
center of the cylinder, stiffening the shell and prevent- 
ing the splitting which would otherwise have occurred. 
In the second place, the fact that the flanges were 
thicker than the shell caused a zone of weakness near 
the flange due to shrinkage in cooling, and the pres- 
ence of what founders call ' ' a hot spot' ' . 

The stresses figured from formula (14) in the cases 
where the failure was on a circumference, are from 
one-fifth to one-sixth the tensile strength of the test 
bar. 

The strength of a chain is the strength of the 
weakest link, and when the tensile stress exceeded the 
strength of the metal near some blow hole or ^*hot 
spot", tearing began there and gradually extended 
around the circumference. 

Values of c as given by equation (a) have been 
calculated for each cylinder, and agree very well except 
in numbers 3 and 5. 

To the criticism that most of the cylinders did not 
fail by splitting, and that therefore formulas (a) and 
(21) are not applicable, the answers would be that the 
chances of failure in the two directions seem about 
equal, and consequently we may regard each cylinder 
as about to fail by splitting under the final pressure. 

If we substitute the average value of c=.05 and a 
safe value of s = 20oo, formula (21) reduces to: 



t=:-Pl-+ JL lp + _P^ (2^) 

8000 200\ ^ 1600 ^ ^^ 

In Kent's Mechanical Engineer's Pocket Book 
p. 794, the following formula is given as representing 
closely existing practice : 

t=.ooo4dp+o.3 inch (24) 

This corresponds to Ban's formula if we take p=i25 
pounds per square inch. 



24 MACHINE DESIGN. 

exampi.es. 

1. Referring to Table VII, verify in at least three 
experiments the values of S and c as there given. 

2. The steam cylinder of a Baldwin locomotive 
is 22 ins. in diameter and 1.25 ins. thick, Assuming 
125 lbs. gauge pressure, find the value of c. Calculate 
thickness by Van Buren's and Barr's formulas. 

3. Determine proper thickness for cylinder of cast 
iron, if the diameter is 38 inches and the steam pres- 
sure 100 lbs. by formulas 13, 20, 21, 23 and 24. 

9. Thickness 6f Flat Plates. An approximate 
formula for the thickness of flat cast-iron plates may 
be derived as follows : 

Let l=length of plate in inches. 
b=breadth of plate in inches. 
t= thickness of plate in inches. 
p=intensity of pressure in pounds. 
S= modulus of rupture lbs. per sq. in. 

Suppose the plate to be divided lengthwise into flat 
strips an inch wide 1 inches long, and suppose that 
a fraction p' of the whole pressure causes the bending 
of these strips. 

Regarding the strips as beams with fixed ends and 
uniformly loaded : 

6M _ 6W1 ^ pT^ 

bh^ ~ i2bh^~ 2t^ 
and the thickness necessary to resist bending is : 



=g^ 



s « 

In a similar manner, if we suppose the plate to be 
divided into transverse strips an inch wide and b inches 
long, and suppose the remainder of the pressure p— p' 
equals p" to cause the bending in this direction, we 
shall have : 

h *' 

But as all these strips form one and the same plate 



-J 



MACHINK DESIGN. 25 

the ratio of p' to p'' must be such that the deflection at 
the center of the plate may be the same on either sup- 
position. The general formula for deflection in this 
case is 

t' 
and 1= — for each set of strips. Therefore the deflec- 
12 

tion IS proportional to -^y~ ^^^ ^~r~ '^ ^^^ ^^^ cases. 

.-. pl^=p''b^ 

But '+p''=P 

Solving in these equations for p^ and p^^ 

p i*+b* 

Substituting these values in (a) and (b) : 



^=^^^>l.-s(iw (^^) 

As 1 > b usually, equation (26) is the one to be 
used. If the plate is square 1 ^^^ b and 



2 \ s 



(27) 



If the plate is merely supported at the edges then 
formulas (25) and (26) become : 
For rectangular plate : 

*-T\i S(P+b*) ^"^> 

For square plate : 



=A |3P 

■ 2\2S 



* — ^J:;o (29) 



25 MACHINE DESIGN. 

Formulas for the thickness of a fiat plate under a 
concentrated load at the center, can be derived in a 
similar manner. A round plate may be treated as 
square, with side = diameter, without sensible error. 

The preceding formulas can only be reparded as 
approximate. Grashof has investigated this subject 
and developed rational formulas but his work is too 
long and complicated for introduction here. His for- 
mulas for round plates are as follows : 

Round plates : 
Supported at edges : 

'=Tji (3°) 

Fixed at edges : 

'=tJ •■•■ •■■■(-) 

where t and p are the same as before, d is the diameter 
in inches and S is the safe tensile strength of the 
material. 

Comparing these formulas with (27) and (29) for 
square plates, they are seen to be nearly identical. 

Experiments made at the Case School of Applied 
Science in 1896-97 on rectangular cast iron plates with 
load concentrated at the center gave results as follows: 
Twelve rectangular plates planed on one side and each 
having an unsupported area of ten by 1 5 inches were 
broken by the application of a circular steel plunger 
one inch in diameter at the geometrical center of each 
plate. The plates varied in thickness from one-half 
inch to one and one-eighth inches. Numbers i to 6 
were merely supported at the edges, while the remain- 
ing six were clamped rigidly at regular intervals around 
the edge. 

To determine the value of S, the modulus of rup- 
ture of the material, pieces were cut from the edge of 
the plates and tested by cross-breaking. The average 
value of S from seven experiments was found to be 
33000 lbs. per sq. in. 



MACHINE DESIGN. 



27 



In Table VIII are given the values obtained for 
the breaking load W under the different conditions. 

Those plates which 
were merely supported at 
the edges broke in three 
or four straight lines rad- 
iating from the center. 
Those fixed at the edges 
broke in four or five rad- 
ial lines meeting an irreg- 
ular oval inscribed in the 
rectangle . Number 1 2 
however failed by shear- 
ing, the circular plunger 
making a circular hole 
in the plate with several 
radial cracks. 

EXAMPLES. 

1 . Calculate the thick- 
ness of a steam-chest 
cover 8X12 inches to 
sustain a pressure of 90 
lbs. per sq. inch with a 
factor of safety = 10. 

2. Calculate the thick- 
ness of a circular man- 
hole cover of cast-iron 18 
inches in diameter to sus- 
tain a pressure of 150 lbs. 
per sq. inch with a factor 
of safety =8, regarding 
the edges as merely sup- 
ported. 

3. Work out formulas for a rectangular plate 
having a concentrated load= W at the center, and with 
edges either supported or fixed. 

4. Test values for W given in Table VIII by 
formulas obtained in example 3. 



TABLE VIII. 


Cast iron plates : 


[OX15 ins. 


No. 


Thick- 
ness. 
t 


Breaking 

Load. 

W 


I 


.562 


7500 


2 


.641 


1 1840 


3 


.745 


14800 


4 


.828 


21900 


5 


1.040 


31200 


6 


I. 120 


31800 


7 


.481 


9800 


8 


.646 


17650 


9 


.769 


26400 


ID 


.881 


33400 


II 


1.020 


47200 


12 


1. 123 


59600 



28 MACHINK DESIGN. 

5. In experiments on steam cylinders, a head 12 
inches in diameter and 1.18 inches thick failed under a 
pressure of 900 lbs. per sq. in. Determine the value 
of S by formula (31). 

10. Machine Frames. For general principles of 
frame design the reader is referred to Chapter 2. Cast 
iron is the material most used but steel castings are 
now becoming common in situations where the stresses 
are unusually great, as in the frames of presses, shears 
and rolls for shaping steel. 

Cored vs. Rib Sections. Formerly the flanged or 
rib section was used almost exclusively, as but a few 
castings were made from each pattern and the cost of 
the latter was a considerable item. Of late years the 
use of hollow sections has become more common; the 
patterns are more durable and more easily molded than 
those having many projections and the frames when 
finished are more pleasing in appearance. 

The first cost of pattern for hollow work, includ- 
ing the cost of the core-box, is sometimes considerably 
more but the pattern is less likely to change its shape 
and in these days of many castings from one pattern, 
this latter point is of more importance. Finally it may 
be said that hollow sections are usually stronger for 
the same weight of metal than any that can be shaped 
from webs and flanges. 

Resistance to Bending. Most machine frames are 
exposed to bending in one or two 
directions. If the section is to 
be ribbed it should be of the form y/ 

shown in Fig. 4. The metal ^ 

being of nearly uniform thickness ^ 

and the flange which is in tension /X 

having an area three or four times 
that of the compression flange. 
In a steel casting these may be 




i 




in a sreei casting tnese may oe (7 //nJ// //? y///y?/ % 
more nearly equal. The hollow V////^<////u//////> 



section may be of the shape shown . 

in Fig. 5, a hollow rectangle with ^^* 4« 



MACHINE DESIGN. 



29 




VZY////////?m 



the tension side re-enforced and slightly thicker than 
the other three sides. The re-enforcing flanges at A 

and B may often be utiUzed 
for the attaching of other 
members to the frame as 
in shapers or drill presses. 
The box section has one 
great advantage over the I 
section in that its moment 
of resistance to side bending 
or to twisting is usually 
B much greater. The double 
Fig. 5. I or the U section is com- 

mon where it is necessary to 
have two parallel ways for sliding pieces as in lathes 
and planers. As is shown in Fig. 6 the two Is are 
i^yyyy J uyy jjy i usually conucctcd at intervals bv 

^T ^ , P^ cross girts. 

Besides making the cross- 
section of the most economical 
form, it is often desirable to have 
such a longitudinal profile as shall 
give a uniform fibre stress from 
end to end. This necessitates a 
parabolic or elUptic outhne of 
which the best instance is the 
housing or upright of a modern 
iron planer. 
Resistance to Twisting;. The 
hollow circular section is the ideal 
form for all frames or machine 
members which are subjected to 
torsion. If subjected also to I 
bending the section may be made 
elliptical or, as is more common, 
thickened on two sides by making 
the core oval. See Fig. 7. As 
has already been pointed out the ^ 

box sections are in general better 
adapted to resist twisting than the ribbed or I sections 



, 



! 



Fig. 6. 




30 MACHINE DESIGN. 

Frames of Machine Tools. The beds of lathes are 
subjected to bending on account of their own weight 
and that of the saddle and on account of the downward 
pressure on the tool when work is being turned. They 
are usually subjected to torsion on account of the un- 
even pressure of the supports. The box section is then 
the best ; the double I commonly used is very weak 
against twisting. The same principle would apply in 
designing the beds of planers but the usual method of 
driving the table by means of a gear and rack prevents 
the use of the box section. The uprights of planers and 
the cross rail are subjected to severe bending moments 
and should have profiles of uniform strength. The up- 
rights are also subject to side bending when the tool 
is taking a heavy side cut near the top. To provide 
for this the uprights may be of a box section or may 
be reinforced by outside ribs. 

The upright of a drill press or vertical shaper is 
exposed to a constant bending moment equal to the 
upward pressure on the cutter X the distance from cen- 
ter of cutter to center of upright. It should then be of 
constant cross-section from the bottom to the top of the 
straight part. The curved or goose-necked portion 
should then taper gradually. 

The frame of a shear press or punch is usually of 
the G shape in profile with the inner fibers in tension 
and the outer in compression. The cross-section should 
be as in Fig. 4 or Fig. 5, preferably the latter and should 
be graduated to the magnitude of the bending moment 
at each point. 

EXERCISES. 

1. Discuss the stresses and the arrangement of 
material in the girder frame of a Corliss engine. 

2. Ditto in the G frame of a band saw. 



II. spiral Springs. The most common form of 
spring used in machinery is the spiral or hehcal spring 
made of round brass or steel wire. Such springs may 
be used to resist extension or compression or they may 
be used to resist a twisting moment. 

Tension and Compression, 
lyCt Iy=length of axis of spring. 
D=mean diameter of spring. 
1= developed length of wire. 
d= diameter of wire. 
n= number of coils. 
P= tensile or compressive force. 
x= corresponding extension or compression. 
S=safe torsional or shearing strength of wire. 
=2500 for spring brass wire. 
= 75000 to 1 15000 for cast steel tempered. 
G= modulus of torsional elasticity. 
= 6000000 for spring brass wire. 
= 12000000 to 15000000 for cast steel, tempered. 

Then l=\/^^^D^n^+I?" 

If the spring were extended until the wire became 
straight it would then be twisted n times, or through an 
angle=27rn and the stretch would be 1 — L. 

The angle of torsion for a stretch =x is then 

^ 27rnx , . 

''=1=17 (^^ 

Suppose that a force P' acting at a radius will 

twist this same piece of wire through an angle d caus- 
ing a stress S at the surface of the wire. Then will 

the distortion of the wire per inch of length be s=— i- 

5.1T 5iP^D .K^ 



32 MACHINE DESIGN. 

^ S I0.2 P^Dl . . 

•'•^=T=-7d^ ^"^ 

In thus twisting the wire the force required will 
vary uniformly from o at the beginning to P^ at the 
end provided the elastic limit is not passed, and the 
average force will be 

P^ P^D<9 

= — The work done is therefore 

2 4 

If the wire is twisted through the same angle by 
the gradual application of the direct pressure P, com- 
pressing or extending the spring the amount x , the 
work done will be 

Px _ P^D^ Px 
— But = — 

2 42 

2Px 
.•.P^= - (d) 

Substituting this value of PMn (c) and solving 
for x: 

_Gd^ 
^~io.2Pl 
Substituting the value of from (a) and again 
solving for x : 

10, 



X 



If we neglect the original obliquity of the wire 
then 1 = -Dn and L=o and equation (e) reduces to 

2,55P1D^ 
"" Gd* ^^^ 

Making the same approximation in equation (d) 
we have P^ = P 

i, e, — a force P will twist the wire through approxi- 
mately the same angle when applied to extend or 
compress the spring, as if applied directly to twist a 
piece of straight wire of the same material with a 

lever arm = — 
2 

This may be easily shown by a model. 



MACHINE DESIGN. 33 

The safe working load may be found by solving- 
for P^ in (b) and remembering that P = P^ 

P=^4^ (33) 

2.55 D ^^^^ 

when S is the safe shearing strength. 

Substituting this value of P in (24) we have for 

the safe deflection : 

IDS , , 

13. Square Wire. The value of the stress for a 
square section is: 

where d is the side of square. 

The distortion at the corners caused by twisting 
through an angle 6 is: 

^ ^d 

Equation (c) then becomes: 
^ 6P'D1 

The three principal equations (32), (33) and (34) 
then reduce to: 

1.5PID' , . 

"=-5d^ (35) 

T3 Sd' , .. 

P=T7^ (36) 

"=05^ (37) 

The square section is not so economical of material 
as the round. 

13. Experiments. Tests made on about 1700 
tempered steel springs at the French Spring Works in 
Pittsburg were reported in 1901 by Mr. R. A French. 
These were all compression springs of round steel and 
were given a permanent set before testing by being 



34 



MACHINE DESIGN. 



closed coil to coil several times. Mr. French as a result 
of the experiments arrives at the following conclusions: 

1. The average value of G is 14,500,000. 

2. The safe working stress S depends upon the 
proportions of the spring and varies from 75000 to 
112,000 lbs. per sq. inch for a good grade of steel 
properly tempered. 

3. If R=--T- the ratio of spring diameter to wire 

diameter, the following values of S may be safely 
assumed. 



VALUES OF S. 




R=3 


R=8 


d = s/s inch or less 

d=-Y^ inch to ^/i inch.. 
d=-Y|-inch to i^ inch... 


112,000 
110,000 
105,000 


85,000 
80,000 
75,000 



4. When a spring is subjected to sudden shocks 
a smaller value of S must be used. 

5. In designing close coil extension springs the 
value of G will be as above but the values of S should 
not be over two-thirds the corresponding values for 
compression springs. 

14. Spring in Torsion. If a spiral spring is used 
to resist torsion instead of tension or compression, the 
wire itself is subjected to a bending moment. We will 
use the same notation as in the last article, only that 
P will be taken as a force acting tangentially to the 

circumference of the spring at a distance — from the 

axis, and S will now be the safe transverse strength 
of the wire, having the following values: 
S=3ooo for vSpring brass wire. 
= 90,000 to 125000 for cast steel tempered. 



MACHINE DKSIGN. 35 

E = 9000000 for Spring brass wire. 

= 30000000 for cast steel tempered. 
Let ^= angle through which the spring is turned 
by P. 

The bending moment on the wire will be the 

PD 
same throughout and = This is best illustrated 

by a model. 

To entirely straighten the wire by unwinding the 
spring would require the same force as to bend straight 
wire to the curvature of the helix. 

To simplify the equations we will disregard the 
obliquity of the helix, then will l=7rDn and the radius 
of curvature _ D 

2 
Let M ^= bending moment caused by entirely 
5>i:raightening the wire ; then by mechanics 

^^ EI 2EI ' 

and the corresponding angle through which spring is 
turned is 27rn. 

But it is assumed that a force P with a radius — 
turns the spring through an angle 0, ^ 

PD 2EI d 

2 D 2-n 

_EI^_EI^ 
■~-Dn~ 1 
Solving for ^ : 

-li <>) 

and if wire is round 

^ 10.2PDI 

'=-^d^ ^38) 

The bending moment for round wire will be 

PD Sd^ . , 

— -irrz (39) 

2 10,2 



3b MACHINK DESIGN. 

and this will also be the safe twisting moment that 
can be applied to the spring when vS = working 
strength of wire. The safe angle of deflection is founa 

PD 

by substituting this value of in (38): 

2IS 
Reducing: ^^^d~ **" ^^^^ 

15. Flat Springs. Ordinary flat springs of uniform 
rectangular cross-section can be treated as beams and 
their strength and deflection calculated by the usual 
formulas. 

In such a spring the bending and the stress are 
greatest at some one point and the curvature is not 
uniform. 

To correct this fault the spring is made of a con- 
stant depth but varying width. 

If the spring is fixed at one end and loaded at the 
other the plan should be a triangle with the apex at 
the loaded end. If it is supported at the two ends and 
loaded at the center, the plan should be two triangles 
with their bases together under the load forming a 
rhombus. The deflection of such a spring is one and 
a half times that of a rectangular spring. 

As such a spring might be of an inconvenient 

width, a com- 
pound or leaf- 
spring is made 
by cutting the 
triangular 
I spr i ng into 

I ^ I -| strips parallel 

I [ to the axis, and 

I piling one a- 
bove another as 
Fig. 8. in Fig. 8. 





f 



This arrangement does not change the principle, 
save that the friction between the leaves may increase 
the resistance somewhat. 



MACHINE DESIGN. 37 

Let l=length of Span. 
b=breadth of leaves, 
t^^thickness of leaves. 
n=: number of leaves. 
W=load at center. 
A=deflection at center. 

S and E may be taken as 80000 and 30000000 
respectively. 



Strength : 



^^ Wl Snbt: 

M= — = 



Elasticity : 



4 6 
w=— -^ — (41) 



WP ^ , nbt' 

A= — =rrwnere I = 

32EI 12 

3WI' , , 

•■• ^=8E^' <42) 

For the benefit of those who wish to design 

springs in quantity, reference is made to Trans. Am. 

Soc. Mech. Eng. Vol. XVII. p. 340, where will be 

found very complete tables for both helical and flat 

springs. 

EXAMPLES. 

1. A spring balance is to weigh 25 pounds with 
an extension of 2 inches, the diameter of spring being 
y^ inches and the material, tempered steel. 

Determine the diameter and length of wire, and 
number of coils. 

2. Determine the safe twisting moment and angle 
of torsion for the spring in example i, if used for a 
torsional spring. 

3. Design a compound flat spring for a locomotive 
to sustain a load of 16000 lbs. at the center, the span 
being 40 inches, the number of leaves 1 2 and the ma- 
terial steel. 

4. Determine the maximum deflection of the 
above spring, under the working load. 



38 MACHINE DESIGN. 

5. Measure various indicator springs and deter- 
mine value of G from rating of springs. 

6. Measure various brass extension springs cal- 
culate safe static load and safe stretch. 

7. Make an experiment on torsion spring to de- 
termine distortion under a given load and calculate 
value of E. 



FASTENINGS. 

i6. Bolts and Nuts. Tables of dimensions for 
U. S. standard bolts and nuts are to be found in all 
hand books and will not be repeated here. 

Roughly the diameter at root of thread is .83 Ox 
the outer diameter in this system, and the pitch in 
inches is given by the formula 

p=.24\/d+.625 — .175 (43> 

where d::= outer diameter. 

In designing bolts to resist simple tension, calcu- 
late the area needed to resist the given tension, divide 
this by the number of bolts to be used and the quotient 
will be the area of one bolt at the root of thread. 

From the tables the corresponding diameter and 
the diameter of body of bolt can be determined. 

Bolts may be divided into three classes which are 
given in the order of their merit. 

1. Through bolts, having a head on one end and 
a nut on the other. 

2. Stud bolts, having a nut on one end and the 
other screwed into the casting. 

3. Tap bolts or screws having a head at one end 
and the other screwed into the casting. 

The principal objection to the last two forms and 
especially to (3) is the liability of sticking or break- 
ing off in the casting. 

Any irregularity in the bearing surfaces of head 
or nut where they come in contact with the casting, 
causes a bending action and consequent danger of 
rupture. 

17. Eye Bolts and Hooks. In designing eye bolts 
it is customary to make the combined sectional area 
of the sides of the eye one and one half-times that of 



4° 



MACHINE DESIGN. 



TABLE IV.-SAFE WORKING STRENGTH 
OF WROUGHT IRON BOLTS. 



Diam 

of 
Bolt. 

Inch. 



1 6 

3/8 

7 

16 

V2 

9 
16 

5/8 
3/ 
/8 
I 



iM 



I 3/^ 

l5/i 
13/ 
I^ 
2 



Diam. 

at 
Root of 
Thread. 
Inches. 



344 
400 

454 
507 
620 

731 

837 
940 

065 

160 

284 

389 
490 

615 
712 



Area 
at 
Root of 
Thread. 
Sq. Ins. 



.0269 

.0452 

.0679 

.0930 

.1257 

.162 

.202 

.302 

.420 

.550 
.694 

.891 

1.057 
1.295 
1. 515 
1.744 
2.049 
2.302 



Safe 
lyoad in 
Tension 

Lbs. 



150 
250 

375 

510 

690 

890 

mo 

1660 

2310 

3025 

3815 
4900 

5815 
7^25 

8335 

9590 

11270 

12660 



Safe 
Load in 
Shear, 

Lbs. 



220 

350 
500 

675 
880 

1120 

1380 

2000 

2700 

3535 

4475 

5520 

6680 

7950 

9330 

10825 

12425 

14130 



Thr'ds 
per 
Inch. 



No. 



20 
18 
16 

14 

13 
12 

II 

10 

9 

8 

7 

7 
6 

6 

5K2 

5 
5 
4/2 



MACHINE DKSIGN. 41 

the bolt to allow for obliquity and an uneven distri- 
bution of stress. 

Large hooks should be designed to resist com- 
bined bending and tension ; the bending moment is 
equal to the load X the longest perpendicular from the 
center line of hook to line of load. 

Check Nuts: A check is a thin nut screwed firmly 
against the main nut to prevent its working loose, and 
is commonly placed outside. 

As the whole load is liable to come on the outer 
nut, it would be more correct to put the main nut out- 
side. 

After both nuts are firmly screwed down, the 
outer one should be held stationary and the inner one 
reversed against it with what force is deemed safe, 
that the greater reaction may be between the nuts. 

The foregoing table is convenient for determining 
the size of bolt needed to resist tension or shear and is 
based on the U. S. standard form of thread using a 
factor of safety =10. 

For steel bolts, increase the loads given in the 
table 20 per cent. The loads given are correct within 
10 pounds. The shearing area used is that of the 
body of the bolt. 

EXAMPLES. 

1. Discuss the effect of the initial tension caused 
by the screwing up of the nut on tb e bolt, in the case 
of steam fittings, etc.; i, e, should this tension be added 
to the tension due to the steam pressure, in determin- 
ing the proper size of bolt ? 

2. Determine the number of Y^ inch bolts neces- 
sary to hold on the head of a steam cylinder 15 inches 
diameter, with the internal pressure 90 pounds per 
square inch, and factor of safety =12. 

3. Show what is the proper angle between the 
handle and the jaws of a fork wrench 

(i) If used for square nuts: 

(2) If used for hexagon nuts; illustrate by figure. 



42 



MACHINE DESIGN. 



4. Determine the length of nnt theoretically 
necessary to prevent stripping of the thread, in terms 
of the outer diameter of the bolt. 

(i) With U. S. standard thread. 

(2) With square thread of same depth. 

5. Design a hook with a swivel and eye at the 
top to hold a load of one ton with a factor of safety = 
5, the center line of hook being three inches from line 
of load, and the material wrought iron. 

18. Riveted Joints. No attempt will be made to 
go into the details of this subject, but only to state 
the general principles involved in designing joints. 

Riveted joints may be divided into two general 
classes : lap joints where the two plates lap over each 
other, and butt joints where the edges of the plates 
butt together and are joined by over-lapping straps or 
welts. If the strap is on one side only, the joint is 
known as a butt joint with one strap; if straps are used 
inside and out the joint is called a butt joint with two 
straps. Butt joints are generally used when the material 
is more than one half inch thick. 

Any joint may have one, two or more rows of 
rivets and hence be known as a single riveted joint, a 
double riveted joint, etc. 

Any riveted joint 

is weaker than the 

original plate, 

nsimply because 

holes cannot be 

punched or drilled 

in the plate for 
the introduction of 
rivets without re- 
moving some of 
the metal. 

Fig. 9 shows a double riveted lap joint and Fig. 
10 a single riveted butt joint with two straps. 

Riveted joints may fail in any one of four Vays : 





c 




c 



i 
i 



) 




) 




MACHINE DESIGN. 43 

1. By tearing of 
the plate along a 
line of rivet holes, 
as at AB, Fig. 9. 

2. By shearing of 
the rivets. 

3. By crushing and 
wrinkling of the 
plate in front of each 
rivet as at C, Fig. 
9, thus causing 
leakage. 

4. By splitting of the plate opposite each rivet as 
at D, Fig. 9. The last manner of failure may be pre- 
vented by having a sufficient distance from the rivet 
to the edge of the plate. Practice has shown that this 
distance should be at least equal to the diameter of a 
rivet. 

Let : t= thickness of plate. 

d= diameter of rivet-hole 

p= pitch of rivets. 

n=number of rows of rivets. 

T= tensile strength of plate. 

C= bearing or crushing strength of plate. 

S= Shearing strength of rivet. 



Average values of the constants are as 


follows : 


Material. 


T 


C 


S 


Wrought Iron. 
Soft Steel. 


50 000 
56 000 


80 000 
90 000 


40 000 
45 000 



19. Lap Joints. This division also includes butt 
joints which have but one strap. 

Let us consider the shell as divided into strips at 
right angles to the seam and each of a width = p . 
Then the forces acting on each strip are the same and 
we need to consider but one strip. 



44 MACHINE DKSIGN. 

The resistance to tearing across of the strip be- 
tween rivet holes is (p— d)tT •••(a) 

and this is independent of the number of rows of 
rivets. 

The resistance to compression in front of rivets is 

ndtC (b) 

and the resistance to shearing of the rivets is 

^nd^S (c) 

The values of the constants given above are only 
average values and are liable to be modified by the 
exact grade of material used and the manner in which 
it is used. 

The tensile strength of soft steel is reduced by 
punching from three to twelve per cent according to 
the kind of punch used and the width of pitch. The 
shearing strength of the rivets is diminished by their 
tendency to tip over or bend if they do not fill the 
holes, while the bearing or compression is doubtless 
relieved to some extent by the friction of the joint. 
The average values given allow roughly for these 
modifications. 

If we call the tensile strength T=unity then the 
relative values of C and S are 1.6 and 0.8 respectively. 

Substituting these relative values of T, C and S 
in our equations, by equating (b) and (c) and reducing 
we have d=2.55t (44) 

Equating (a) and (c) and reducing we have 

p=d+.628-^ (45) 

Or by equating (a) and (b) 

p=d+i.6nd (46) 

These proportions will give a joint of equal 
strength throughout, for the values of constants as- 
sumed. 

20. Butt Joints with two Straps. In this case the 
resistance to shearing is increased by the fact that the 



MACHINE DESIGN. 45 

rivets must be sheared at both ends before the joint 
can give way. Experiment has shown this increase 
of shearing strength to be about 85 per cent and we 
can therefore take the relative value of S as 1.5 for 
butt joints. 

This gives the following values for d and p 

d=i.36t .^ (47) 

p=d+i.i8^ (48) 

p=d+i.6nd (49) 

In the preceding formulas the diameter of hole 
and rivet have been assumed to be the same. 

The diameter of the cold rivet before insertion 
will be -^\ inches less than the diameter given by the 
formulas. 

Experiments made in England by Prof. Kennedy 
give the following as the proportions of maximum 
strength : 

lyap joints d = 2.33t 

p=d+i.375nd 

Butt joints d=i.8t 

p=d+r.55nd 

21. Efficiency of Joints. The efl&ciency of joints 
designed like the preceding is simply the ratio of the 
section of plate left between the rivets to the section 
of solid plate, or the ratio of the clear distance between 
two adjacent rivet holes to the pitch. From formula 
(35) we thus have 

Eflficiency= — -^— ^ — (50) 

i + i.6n 

This gives the efficiency of single, double and 

triple riveted seams as 

.615, .762 and .828 respectively. 

Notice that the advantage of a double or triple 
riveted seam over the single is in the fact that the pitch 
bears a greater ratio to the diameter of a rivet, and 
therefore the proportion of metal removed is less. 



4^ machine: design. 

22. Butt Joints with unequal Straps. One joint 
in common use requires special treatment. 



It is 
the inner 



?? 

fi 




c 



1 



c 



i 



) 
) 




a double - riveted butt joint in which 
strap is made wider than the outer 

and an 

extra row 

of rivets 

added, 

whose 

pitch is 

double 

that of 
the origi- 
nal seam ; 

this is 

sometimes 

called 

diamond 

riveting. 

See Fig. 
II. 




Fig. II. 

This outer row of rivets is then exposed to single 
shear and the original rows to double shear. 

Consider a strip of plate of a widths 2 p. Then 
the resistance to tearing along the outer row of rivets 
is (2p— d)tT 

As there are five rivets to compress in this strip 
the bearing resistance is 

5dtC 

As there is one rivet in single shear and four in 
double shear the resistance to shearing is 

I i + (4X 1.85) J — d^S=6.6d^S 



MACHINE DESIGN. 47 

Solving these equations as in previous cases, we 
have for this particular joint 

d=i.52t (51) 

2p=9d 
P=4-5d .-(52) 

Efficiency=-^ = — (53) 

2p 9 

23. Practical Rules. The formulas given above 

show the proportions of the usual forms of joints for 

uniform strength. 

In practice certain modifications are made for 
economic reasons. To avoid great variation in the 
sizes of rivets the latter are graded by sixteenths of an 
inch, making those for the thicker plates considerably 
smaller than the formula would allow, and the pitch 
is then calculated to give equal tearing and shearing 
strength. 

The following table gives the proportions gener- 
ally used in this country for lap joints, as given by 
'^Locomotive'' 1882. 



TABLE X.-RIVETED LAP JOINTS. 


Thick- 
ness of 


Diam. 
of 


Diam. 
of 


Pitch. 


Efficiency. 










Plate. 


Rivet. 


Hole. 


Single. 


Double 


Single. 


Double 


% 


^8 


1 1 
16 


2 


3 


.66 


•77 


5 
1 6 


1 1 

16 


^ 


2tV 


zyi 


.64 


.76 


3/8 


Va 


1 3 

1 6 


2yi 


z% 


.62 


•75 


16 


1 3 

16 


^8 


2-A- 


33/8 


.60 


•74 


y- 


^8 


15 
16 


^% 


1% 


•58 


•73 



This table is for iron plates and iron rivets. For 
steel plates with iron or steel rivets increase the diam- 
eter of rivets -^ inch, the pitch remaining the same. 



48 



MACHINE DESIGN. 



A similar table has been calculated for butt joints. 
Table XI is for iron plates with iron rivets. For steel 
plates increase the diameter of rivet -jj- inch, the pitch 
remaining the same. 



TABLE XI.- 


-RIVETED BUTT JOINTS. 


Thick- 
ness of 


Diam. 
of 


Diam. 
of 


Pitch. 








Plate. 


Rivet. 


Hole. 


Single. 


Double 


Triple. 


V2 


Va 


13 

16 


2Y& 


4 


s'A 


Vz 


13 

16 


/8 


2 3/^ 


zV. 


sY. 


Va 


n 


15 
16 


2 3/^ 


2,Va 


sH 


% 


15 
16 


I 


2 Yd 


2>Va 


5 


I 


I 


i-iV 


2 3/^ 


3K 


5 



EXAMPIvES. 

1. Investigate proportions of joints for half -inch 
plate in Table X and criticise. 

2. Criticise in same way the proportions of joints 
for one inch plate in Table XI. 

3. Show the effect of increasing the diameter of 
rivets -jj- inch for steel plates and prove by example. 

4. A cylindrical boiler 5X 16 ft. is to have long 
seams double-riveted laps and ring seams single riveted 
laps. If the internal pressure is 90 lbs. gauge pres- 
sure and the material soft steel, determine thickness 
of plate and proportions of joints. Factor of safety to 
be five and efficiency of joints to be allowed for. 

5. A marine boiler is 11 ft. 6 ins. in diameter and 
14 ft. long. The long seams are to be diamond riveted 
butt joints and the ring seams ordinary double riveted 
butt joints. The internal pressure is to be 175 lbs. 
gauge and the material is to be steel of 60,000 lbs. 



MACHINE DESIGN. 



49 



tensile strength. Determine thickness of shell and 
proportions of joints. Net factor of safety to be five 
allowing for efiiciency of joints. 

6. Design a diamond riveted joint such as shown 
in Fig. iia for a steel plate f^ inches thick. Outer 
cover plate is 5^ inches and inner cover plate tV inches 
thick. Determine efficiency of joints. 




Fig. 1 1 a. 

7. The single lap joint with cover plate, as shown 
in Fig. 12, is to have pitch of outer rivets double that 
of inner row. Determine diameter and pitch of rivets 
for fi inch plate and the efficiency of joint. 



\_7 



^ 



Fig. 12 



14. Joint Pins. A joint pin is a bolt exposed to 
double shear. If the pin is loose in its bearings it 
should be designed with allowance for bending, by 
adding from 30 to 50 per cent to the area of cross- 
section needed to resist shearing alone. Bending of the 
pin also tends to spread apart the bearings and this 
should be prevented by having a head and nut or cot- 
ter on the pin. 

If the pin is used to connect a knuckle joint as in 
boiler stays, the eyes forming the joint should have a 
a net area 50 per cent in excess of the body of the 
stay, to allow for bending and uneven tension. 



50 MACHINE DESIGN. 

25. Cotters. A cotter is a key which passes dia- 
metrically through a- hub and its rod or shaft, to fasten 
them together, and is so called to distinguish it from 
shafting keys which lie parallel to axis of shaft. 

Its taper should not be more than 4 degrees or 
about I in 15, unless it is secured by a screw or check 
nut. 

The rod is sometimes enlarged where it goes in 
the hub, so that the effective area of cross-section 
where the cotter goes through may be the same as in 
the body of the rod 

Let: d=diameter of body of rod. 

d^^^diameter of enlarged portion. 

t=thickness of cotter, usually = — - 

4 
b= breadth of cotter. 

1 = length of rod beyond cotter. 
Suppose that the applied force is a pull on the rod 
— causing tension on the rod and shearing stress on 
the cotter. 

The effective area of cross section of rod at cotter 

is ^Ail^^ ^._.dy 

4 4 ^^ 4 

and this should equal the area of cross-section of the 
body of rod. 

4 4 

^^=^Jr^, = i.2id (54) 

Let P=pull on rod. 

S=shearing strength of material. 
The area to resist shearing of cotter is 

2 S 
. K 2P 
•• ''^^ • (^> 



MACHINE DKSIGN. 

The area to resist shearing of rod is 



51 



and 1= 



2d,S 



(b) 



If the metal of rod and cotter are the same 

2d,l = ^A 



1 



(55) 



Great care should be taken in fitting cotters that 
they may not bear on corners of hole and thus tear the 
rod in two. 

A cotter or pin subjected to alternate stresses in 
opposite directions should have a factor of safety 
double that otherwise allowed. 

Adjustable cotters, used 
for tightening joints or bear- 
ings are usually accompanied 
by a gib having a taper equal 
and opposite to that of the 
cotter. (Fig. 13) In design- 
ing these for strength the 
two can be regarded as re- 
sisting shear together. 

For shafting keys see 
chapter on shafting. 

EXAMPI^ES. 

1. Design a knuckle joint for a soft steel boiler 
stay, the pull on stay being 12000 lbs. and the factor 
of safety, six. 

2. Determine the diameter of a round cotter pin 
for equal strength of rod and pin 

3. A rod of wrought iron has keyed to it a piston 
18 inches in diameter, by a cotter of machinery steel. 




52 MACHINE DESIGN. 

Required the two diameters of rod and dimensions 
of cotter to sustain a pressure of 1 50 pounds per square 
inch on the piston. Factor of safety = 8. 

4. Design a cotter and gib for connecting rod of 
engine mentioned in Ex. 3, both to be of machinery 
steel and ,75 inches thick. 



SLIDING BEARINGS^ 

26. Slides in General. The surfaces of all slides 
should have sufficient area to limit the intensity of 
pressure and prevent forcing out of the lubricant. No 
general rule can be given for the Hmit of pressure. 
Tool marks parallel to the sliding motion should not 
be allowed, as they tend to start grooving. The 
sliding piece should be as long as practicable to avoid 
local wear on stationary piece and for the same reason 
should have sufficient stiffness to prevent springing. 
A slide which is in continuous motion should lap over 
the guides at the ends of stroke, to prevent the wear- 
ing of shoulders on the latter and the finished surfaces 
of all slides should have exactly the same width as the 
surfaces on which they move for a similar reason. 

Where there are two parallel guides to motion as 
in a lathe or planer it is better to have but one of these 
depended upon as an accurate guide and to use the 
other merely as a support. It must be remembered that 
any sliding bearing is but a copy of the ways of the 
machine on which it was planed or ground and in turn 
may reproduce these same errors in other machines. 
The interposition of handscraping is the only cure for 
these hereditary complaints. 

In designing a slide one must consider whether it 
is accuracy of motion that is sought, as in the ways of 
a planer or lathe, or accuracy of position as in the 
head of a milling machine. Slides may be divided ac- 
cording to their shapes into angular, flat and circular 
slides. 

27. Angular Slides, An angular slide is one in 
which the guiding surface is not normal to the direc- 
tion of pressure. There is a tendency to displacement 
sideways, which necessitates a second guiding surface 
inclined to the first. This oblique pressure constitutes 



54 



MACHINK DKSIGN. 



the principal disadvantage of angular slides. Their 
principal advantage is the fact that they are either self 
adjusting for wear, as in the ways of lathes and 
planers, or require at most but one adjustment. 

Fig. 14 shows one of the V^s of an ordinary planing 
machine. The platen is held in place by gravity. 
The angle between the two surfaces is usually 90° but 
may be more in heavy machines. The grooves g, g are 
intended to hold the oil in 
place ; oiling is sometimes 
effected by small rolls re- 
cessed into the lower piece 
and held against the platen 
by springs. 

The principal advantage of 
this form of way is its abil- 
ity to hold oil and the great 
disadvantage its faculty for 
catching chips and dirt. 

/Fig. 15 shows an inverted V such as is com- 
mon on the ways of engine lathes. The angle is about 
the same as in the preceding form but the top of the V 
should be rounded as a precaution against nicks and 
bruises. 

The inverted V is pre- 
ferred for lathes since it 
will not catch dirf and 
chips. It needs frequent 
lubrication as the oil runs 
off rapidly. Some lathe 
carriages are provided 
with extensions filled 
with oily felt or waste to 
protect the ways from 
dirt and keep them wiped and oiled. Side pressure 
tends to throw the carriage from the ways; this action 
may be prevented by a heavy weight hung on the car- 
riage or by gibbing the carriage at the back (See Fig. 20) . 





Fig. 15 



MACHINE DESIGN. 55 

The objection to this latter form of construction is the 
fact that it is practically impossible to make and k^p 
the two V' s and the gibbed slide all parallel. 

28. Gibbed Slides. All shdes which are not 
self-adjusting for wear must be provided with gibs 
and adjusting screws. Fig. 16 shows the most com- 
mon form as used in tool shdes for lathes and planing 
machines. 

The angle employed is 

usually 60°; notice that 

^ Ss -pL the corners c c are clip- 

\ /A'^^"^- 3 ped for strength and to 



g" 



avoid a corner bearing; 
notice also the shape of 
gib. It is better to have 
p. j^ the points of screws 

^' ' coned to fit gib and nol 

to have flat points fitting recesses in gib. The latter 
form tends to spread joint apart by forcing gib down. 
If the gib is too thin it will spring under the screws and 
cause uneven wear. 

The cast iron gib, Fig. 
17, is free from this latter 
defect but makes the slide 
rather clumsy. The screws 
however are more accessi- 
ble in this form. Gibs are 
sometimes made slightly ta 
pering and adjusted by a screw Fig. 17. 

and nut giving endwise motion. 

29. Flat Slides. This type of slide requires ad- 
justment in two directions and is usually provided 
with gibs and adjusting screws. Flat ways on ma- 
chine tools are the rule in English practice and are 
gradually coming into use in this country. Although 
more expensive at first and not so simple they are 
more durable and usually more accurate than the an- 
gular ways. 





— ?ri 

1 t 


\ 


M 



MACHINK DESIGN. 



a flat way for a planing ma- 

_5 




S6 

Fig. 1 8 illustrates 
chine. The other 
way would be simi- 
lar to this but with- 
out adjustment. 
The normal pres- 
sure and the fric- 
tion are less than 
with angular 
ways and no 
amount of side 
pressure will lift 
the platen from 
its position. 

Fig. 19 shows a portion of the ram of a shaping 
machine and illustrates the use of an L gib for adjust- 

^_^ . ,, — . ment in two di- 

llP , I rections. Fig. 20 

shows a g i b b e d 
slide for holding 
down the back of a 
lathe carriage with 
two adjustments. 

The gib g is 
tapered and adjust- 
ed by a screw and 
nuts. The saddle of 
a planing machine 
the table of a shaper usually has a rectangu- 
lar gibbed slide above and a 
taper slide below, this form 
of the upper slide being ne- 
cessary to hold the weight 
of the overhanging metal. 
Some lathes and planers are 
built with one V or angular 
way for guiding the carriage j 
or platen and one flat wa}^ ^ 
acting merely as a support. 




or 



u 



9) 



9 



TXJ 



Fig. 20. 



MACHINE DKSIGN. 



57 



30. Circular Guides. Examples of this form may 
be found in the column of the drill press and the over- 
hanging arm of the milling machine. The cross heads 
of steam engines are sometimes fitted with circular 
guides; they are more frequently flat or angular. One 
advantage of the circular form is the fact that the cross 
head can adjust itself to bring the wrist pin parallel to 
the crank pin. The guides can be bored at the same 
setting as the cylinder in small engines and thus se- 
cure good alignment. 

31, Stuffing Boxes. In steam engines and pumps 
the glands for holding the steam and water packing 
are the sliding bearings which cause the greatest fric- 
tion and the most trouble. Fig. 2 1 shows the general 

arrangement. 
B is the stufl&ng 
box attached to 

the cylinder 
head; R is the 
piston rod ; G ^ 

the gland ad- 4 
justed by nuts 

on the studs 
shown ; P the 

packing con- 
tained in a re- 
cess in the box 
and consisting 
of rings, either of some elastic fibrous material like 
hemp and woven rubber cloth or of some soft metal 
like babbit. The pressure between the packing and 
the rod, necessary to prevent leakage of steam or wa- 
ter, is the cause of considerable friction and lost work. 
Experiments made from time to time in the labora- 
tories of the Case School of Applied Science have 
shown the extent and manner of variation of this fric- 
tion. The results for steam packings may be sum- 
marized as follows : 

I. That the softer rubber and graphite packings, 




Fig. 21 



58 MACHINE DESIGN. 

which are self-adjusting and self-lubricating, as inNos. 
2, 3, 7, 8, and 1 1, consume less power than the harder 
varieties. No. 17, the old braided flax style, gave very 
good results, 

2. That oiling the rod will reduce the friction with 
any packing. 

3. That there is almost no limit to the loss caused 
by the injudicious use of the monkey-wrench. 

4. That the power loss varies almost directly with 
the steam pressure in the harder varieties, while it is 
approximately constant with the softer kinds. 

The diameter of rod used — two inches— would be 
appropriate for engines from 50 to 100 horse-power. 
The piston speed was about 140 feet per minute in the 
experiments, and the horse-power varied from .036 to 
.400 at 50 pounds steam pressure, with a safe average 
for the softer class of packings of .07 horse-power. 

At a piston speed of 600 feet per minute, the same 
friction would give a loss of from .154 to 1.71 with a 
working average of .30 horse-power, at a mean steam 
pressure of 50 pounds. 

In Table 12 Nos. 6, 14, 15 and 16 are square, 
hard rubber packings without lubricants. 

Similar experiments on hydraulic packings under 
a water pressure varying from ten to eighty pounds 
per square inch gave results as shown in Table 14. 

The figures given are for a two inch rod running 
at an overage piston speed of 140 feet per minute. 



machine; design. 



59 



TABLE XII. 



^ be 


1—i 




^g^g 


, , 4> 




o q 


'u 


H S-^ 


oUr^pq 


^^^ 






6 






(A en . 


Remarks on Leakage, etc. 


I 


5 


22 


.091 


.085 


Moderate leakage. 


2 


8 


40 


.049 


.048 


Easily adjusted; slight leakage. 


3 


5 


25 


•037 


.036 


Considerable leakage. 


4 


5 


25 


•159 


.176 


lycaked badly. 


5 


5 


25 


.095 


.081 


Oiling necessary; leaked badly. 


6 


5 


25 


.368 


.400 


Moderate leakage. 


7 


5 


25 


.067 


.067 


Easily adjusted and no Tkage. 


8 


5 


25 


.082 


.082 


Very satisfactory; slight I'kage. 


9 


3 


15 


.200 


.182 


Moderate leakage. 


lO 


3 




.275 


. 


Excessive leakage. 


II 


5 


25 


.157 


.172 


Moderate leakage. 


12 


5 


25 


.266 


•330 


Moderate leakage. 


13 


5 


25 


.162 


.230 


No leakage; oiling necessary. 


14 


5 


25 


.176 


.276 


Moderate I'kage; oiling neces. 


15 


5 


25 


•233 


.255 


Difficult to adjust; no leakage. 


i6 


5 


25 


.292 


.210 


Oiling necessary; no leakage. 


17 


5 


25 


.128 


.084 


No leakage. 



TABLE XIII. 



Kind 
of 

Pack- 
ing. 


Horse Power consumed by each Box, 
when Pressure was applied to Gland 
Nuts by a Seven - Inch Wrench. 


Horse Power 

before and 

after oiling 

Rod. 




5 

Pounds 


8 

Pounds 


10 

Pounds 


12 

Pounds 


14 

Pounds 


16 
Pounds 


Dry. 


Oiled. 


I 

3 
4 
5 
6 

9 

II 
12 

13 
15 
16 

17 


.120 


.248 
.220 
.348 
.126 

.363 
.6b6 

.405 
.161 

.317 
.526 

.327 

.198 


.136 

•430 
.228 
.500 

•454 
.242 

•394 

*.*86o 

.277 


•303 

.260 
.535 

•359 
.582 

.380 


•330 
.520 

.454 


•390 

.340 
•533 


.055 
.154 

.323 
.067 

•533 
.666 

.454 
.454 


.021 

.123 

.194 

•053 
,236 
.636 
.176 
.122 



6o 


MACHINE 


DESIGN 


• 






TABLE XiV. 






No. of 
Packing. 


Av. H. P. 

at 
2o Lbs. 


Av. H. P. 

at 

70 Lbs. 


Max. 
H. P. 


Min. 
H. P. 


Av. H. P. 

for entire 

Test. 


I 


.077 


.351 


•452 


.024 


■259 


2 


.422 


.500 




512 


.167 


.410 


3 


.130 


.178 




276 


•035 


.120 


4 


.184 


.195 




230 


.142 


.188 


5 


.146 


.162 




285 


.069 


.158 


6 


.240 


.200 




255 


.071 


.186 


7 


.127 


.192 




213 


•095 


•154 


8 


.153 


.174 




238 


.112 


.165 


9 


.287 


.469 




535 


•159 


•389 


lO 


.151 


.160 




226 


•035 


.103 


^ II 


.141 


156 




380 


.064 


.177 


12 


.053 


•095 




•143 


•035 


.090 



Packings Nos. 5, 6, 10 and 12 are braided flax 
with graphite lubrication and are best adapted for low 
pressures. Packings Nos. 3, 4 and 7 are similar to the 
above but have parafine lubrication. Packings Nos. 
2 and 9 are square duck without lubricant and are on- 
ly suitable for very high pressures, the friction loss 
being approximately constant. 

EXAMPLES. 
Make a careful study and sketch of the sliding 
bearings on each of the following machines and ana- 
lyze as to: (a) Purpose (b) Character, (c) Adjust- 
ment, (d) Lubrication . 

1 . One of the engine lathes in the shop. 

2. One of the planing machines. 

3. One of the shaping machines. 

4. One of the milling machines. 

5. One of the upright drills. 

6. One of the engines. 



®Jtai:tt^r 7* 



JOURNALS, PIVOTS AND BEARINGS. 

32. Journals. A journal is that part of a rotating 
shaft which rests in the bearings and is of necessity a 
surface of revolution, usually cylindrical or conical. 
The material of the journal is generally steel, some- 
times soft and sometimes hardened and ground. 

The material of the bearing should be softer than 
the journal and of such a quality as to hold oil readily. 
The cast metals such as cast iron, bronze and babbitt 
metal are suitable on account of their porous, granu- 
lar character. Wood, having the grain normal to the 
bearing surface, is used where water is the lubricant, 
as in water wheel steps and stern bearings of propel- 
lers, 

33. Adjustment. Bearings wear more or less rap- 
idly with use and need to be adjusted to compensate 
for the wear. The adjustment must be of such a char« 
acter and in such a direction as to take up the wear 
and at the same time maintain as far as possible the 
correct shape of the bearing. The adjustment should 
then be in the line of the greatest pressure. 

Fig. 22 illustrates 
some of the more com- 
mon ways of adjusting 
a bearing, the arrows 
showing the direction 
of adjustment and pre- 
sumably the direction 
of pressure, (a) is the 
most usual where the 
principal wear is ver- 
tical, (d) is a form 
frequently used on 
the main journals of 




62 



MACHINE DKSIGisr. 



engines when the wear is in two directions, horizontal 
on account of the steam pressure and vertical on ac- 
count of the weight of shaft and fly wheel. All of 
these are more or less imperfect since the bearing, af- 
ter wear and adjustment, is no longer cylindrical but 
is made up of two or more approximately cylindrical 
surfaces. 

A bearing slightly conical and adjusted endwise 
as it wears, is probably the closest approximation to 
correct practice. 

Fig. 23 shows the 
main bearing of the 
Porter - Allen en- 
gine, one of the 
best examples of a 
four part adjust- 
ment. The cap, is 
adjusted in the nor- 
mal way with bolts 
and nuts; the 
bottom, can be 
raised and lowered 
the cheeks can be 




Fig. 



23. 
underneath 



by liners placed 
moved in or out by means of the wedges shown. 
Thus it is possible, not only to adjust the bearing for 
wear, but to align the shaft perfectly. 

The main bearing of the spindle in a lathe, as 
shown in Fig. 24, offers a good example of symmetri- 
cal adjustment. The headstock A has a conical hole 
to receive the bearing B, 
which latter can be moved 
lengthwise by the nuts F G. 
The bearing may be split 
into two, three or four seg- 
ments or it may be cut as 
shown in (e) Fig. 22 and 
sprung into adjustment. A 
careful distinction must be 
made between this class of 
bearing and that before 




MACHINE DESIGN, 63 

mentioned, where the journal itself is conical and ad- 
justed endwise. A good example of the latter form is 
seen in the spindles of many milling machines. 

34 Lubrication. The bearings of machines which 
run intermittently, like most machine tools, are oiled 
by means of simple oil holes, but machinery which is 
in continuous motion as is the case with line shafting 
and engines requires some automatic system of lu- 
brication. There is not space in these notes for a de- 
tailed description of all the various types of oiling de- 
vices and only a general classification will be at- 
tempted. 

lyubrication is effected in the following ways: 

1 . By grease cups. 

2. By oil cups. 

3. By oily pads of felt or waste. 

4. By oil wells with rings or chains for lifting 
the oil. 

5. By centrifugal force through a hole in the 
journal itself. 

Grease cups have little to recommend them except 
as auxiliary safety devices. Oil cups are various in 
their shapes and methods of operation and constitute 
the chief class of lubricating devices. They may be 
divided according to their operation into wick oilers, 
needle feed, and sight feed. The two first mentioned 
are nearly obsolete and the sight feed oil cup, which 
drops the oil at regular intervals through a glass tube 
in plain sight, is in common use. The best sight feed 
oiler is that which can be readily adjusted as to time 
intervals, which can be turned on or off without dis- 
turbing the adjustment and which shows clearly by its 
appearance whether it is turned on. On engines and 
electric machinery which is in continuous use day and 
night, it is very important that the oiler itself should 
be stationary, so that it may be filled without stopping 
the machinery. 

For continuous oiling of stationary bearings as in 
line shafting and electric machinery, an oil well below 



64 



MACHINE DESIG 



the bearing is preferred, with some automatic means 
of pumping the oil over the bearing, when it runs 
back by gravity into the well. Porous wicks and pads 
acting by capillary attraction are un- 
certain in their action and liable to 
become clogged. For bearings of 
medium size, one or more light steel 
rings running loose on the shaft 
and dipping into the oil, as shown 
in Fig. 25, are the best. For large 
bearings flexible chains are employed 
which take up less room than the ring. 
Centrifugal oilers are most used on 
parts which cannot readily be oiled 
when in motion, such as loose pulleys and the crank 
pins of engines. 

Fig. 26 shows two such devices as applied to an 
engine. In A the oil is supplied by the waste from the 
main journal; in B an external sight feed oil cup is 
used which supplies oil to the central revolving cup C. 



^ 




Fig. 25 








;L... . 


B 













c 



Fig. 26. 



Ivoose pulleys or pulleys running on stationary 
studs are best oiled from a hole running along the axis 
of the shaft and thence out radially to the surface of 
the bearing. A loose bushing of some soft metal per- 
forated with holes is a good safety device for loose 
pulleys. 

Note: For adjustable pedestal and hanger bearings 
see the chapter on shafting. 



MACHINK DESIGN. 65 

35. Friction of Journals: 

Let W=the total load on a journal in lbs. 
l=the length of journal in inches. 
d=the diameter of journal in inches. 
N= number of revolutions per minute. 
v= velocity of rubbing in feet per minute. 
F= friction at surface of journal in lbs. 
= W tan (p nearly. 
If a journal is properly fitted in its bearing and 
does not bind, the value of F will not exceed W tan <p 
and may be slightly less. The value of tan ^ varies ac- 
cording to the materials used and the kind of lubrica- 
tion, from .05 to .01 or even less. The work absorbed 
in friction may be thus expressed : 

^ _^^^ TrdN TrdNWtans^ , ^. 

Fv=Wtan^x = (56) 

12 12 

36. Limits of Pressure. Too great an intensity of 
pressure between the surface of a journal and its bear- 
ing will force out the lubricant and cause heating and 
possibly ''siezing". The safe limit of pressure de- 
pends on the kind of lubricant, the manner of its ap- 
plication and upon whether the pressure is continuous 
or intermittent. The projected area of a journal, or 
the product of its length by its diameter, is used as a 
divisor. 

The journals of railway cars offer a good example 
of continuous pressure and severe service. A limit of 
300 lbs. per square inch of projected area has been 
generally adopted in such cases. 

In the crank and wrist pins of engines, the rever- 
sal of pressure diminishes the chances of the lubricant 
being squeezed out, and a pressure of 500 lbs. per sq. 
in. is generally allowed. 

The use of heavy oils or of an oil bath, and the 
employment of harder materials for the journal and 
its bearing allow of even greater pressures. 

37. Heating of Journals. The proper length of 
journals depends on the liability of heating. 



66 MACHINE DESIGN. 

The energy or work expended in overcoming fric- 
tion is converted into heat and must be conveyed away 
by the material of the rubbing surfaces. If the ratio 
of this energy to the area of the surface exceeds a cer- 
tain Hmit, depending on circumstances, the heat will 
not be conveyed away with sufficient rapidity and the 
bearing will heat. 

The area of the rubbing surface is proportional to 
the projected area or product of the length and diame- 
ter of the journal, and it is this latter area which is 
used in calculation. 

Adopting the same notation as is used in Art. 35, 
we have from equation (56) 

the work of friction = . ft. lbs, 

12 

or = :7rdNWtan ^ inch lbs. 
The work per square inch of projected area is 
then: TidNWtanc^' TrNWtan^ , . 

^= — id — = — I — ^^> 

Solving in (a) for 1 

TcNWtan^ ... 

1= (b) 

w 

w 

Let— =:C a co-efficient whose value is to be ob- 

TTtan^ 

tained by experiment; then 

C = -^andl-=-^ (57) 

Crank pins of steam engines have perhaps caused 
more trouble by heating than any other form of jour- 
nal. A comparison of eight different classes of propel- 
lers in the old U. S. Navy showed an average value 
for C of 350000. 

A similar average for the crank pins of thirteen 
screw steamers in the French Navy gave C= 400000. 

Locomotive crank pins which are in rapid motion 
through the cool outside air allow a much larger value 
of C, sometimes more than a million. 



MACHINE DKSIGN. 67 

Examination of ten modern stationary engines 
shows an average value of C= 200000 and an average 
pressure per square inch of projected area =300 lbs. 

In general we may use these formulas for station- 
ary practice : 

To prevent heating: 1= (s8) 

^ ^ 200000 ^^ ^ 

W 
To prevent wear ld= (59) 

38. Strength and Stiffness of Journals. A jour- 
nal is usually in the condition of a bracket with a uni- 

Wl 
form load, and the bending moment M = — 

Therefore by formula (6) 

Wl 



" --^T- w^ 



A 3 fWl ,^ , 

or d=i.72i' 1-^ (60) 

The maximum deflection of such a bracket is 



/\ = 



8EI 

;rd' WV 



64 8EA 
64WP ^ 2.547WP 

sttEa" ea 

If as is usual A is allowed to be ih inches, then 

for stiffness d=^ J?^^^^ (6i) 

or approximately d=4 ^=:- (6.?) 



68 MACHINE DESIGN. 

The designer must be guided by circumstances in 
determining whether the journal shall be calculated 
for wear, for strength or for stiffness. A safe way is 
to use all three of the formulas and take the largest 
result. 

Remember that no factor of safety, is needed in 
formula for stiffness. 

Note that W in formulas for strength and stiffness 
is not the average but the maximum load. 

39. Caps and Bolts. The cap of a journal bearing 
is in the condition of a beam supported by the holding 
down bolts and loaded at the center, and may be de- 
signed either for strength or for stiffness. 

Let : P=max. upward pressure on cap. 
L = distance between bolts. 
b= breadth of cap at center. 
h=depth of cap at center. 
A=greatest allowable deflection. 

Qf .;. AT Sbh^ PL 
Strength: M = — - — == — 



Stiffness : A = 



-J 



2bS 
WL' 



(63) 



48EI 
_ bh^_ WL^ 



12 48EA 



'--'^^^r^ (64) 



WL' 

4bEA 

If A is allowed to be xio inches and E for cast 
iron is taken = 18000000- 

then: h = .oiii5L'J^ (65) 

The holding down bolts should be so designed 
that the bolts on one side of the cap may be capable 
of carrying safely two thirds of the total pressure. 



MACHINE DESIGN. 69 

exampi.es. 

1. A flat car weighs 10 tons, is designed to carry 
a load of 20 tons more and is supported by two four 
wheeled trucks, the axle journals being of wrought 
iron and the wheels 33 inches in diameter. 

Design the journals, considering heating, wear, 
strength and stiffness, assuming a maximum speed of 
30 miles an hour, factor of safety=io and 0=^300000. 

2. Measure the crank pin of any modern engine 
which is accessible, calculate the various constants and 
compare them with those given in this section. 

3. Design a crank pin for an engine under the 
following conditions : 

Diameter of piston = 28 inches. 

Maximum steam pressure = 90 lbs. per sq. in. 
Mean steam pressure = 40 lbs. per sq. in. 
Revolutions per minute = 75 

Determine dimensions necessary to prevent wear 
and heating and then test for strength and stiffness. 

4. Make a careful study and sketch of journals 
and journal bearings on each of the following 
machines and analyze as to (a) Materials, (b) Ad- 
justment, (c) lyubrication. 

(i) One of the engine lathes in the shop. 

(2) One of the milling machines. 

(3) One of the steam engines. 

(4) One of the electric generators. 

5. Sketch at least one form of oil cup used in the 
laboratories and explain its working. 

6. The shaft journal of a vertical engine is 4 ins. 
in diameter by 6 ins. long. The cap is of cast iron, 
held down by 4 bolts of wTought iron, each 5 ins. 
from center of shaft, and the greatest vertical pressure 
is 12000 lbs. 

Calculate depth of cap at center for both strength 
and stiffness, and also the diameter of bolts. 

7. Investigate the strength of the cap and bolts of 



70 MACHINE DESIGN. 

some pillow block whose dimensions are known, under 
a pressure of 500 lbs. per sq. in. of projected area. 

8. The total weight on the drivers of a locomotive 
is 64000 lbs. The drivers are four in number, 5 ft. 2 
in. in diameter, and have journals 7}^ in. in diameter. 

Determine the horse power consumed in friction 
under each of the three above named conditions, when 
the locomotive is running 50 miles an hour, assuming 
tan^=:i.o5 . 

40. Friction of Pivots or Step -Bearings.— Flat 
Pivots. 

Let W=weight on pivot 

dj=outer diameter of pivot 
p= intensity of vertical pressure 
M = moment of friction 
f= co-efficient of friction = tan ^ 

We will assume p to be a constant which is no 
doubt approximately true. 

W 4W 



Then p= 



area TrdJ 



Let r= the radius of any elementary ring of a 
width = dr , then area of element=27rrdr 

Friction on element = fp X 27rrdr 
Moment of friction of element =2fp7rrMr 

-.1 

and M = 2fp7r f— ^'^^ ^^) 

o 

r^ d' 

or M = 2fp7r =2fp7r — ^ 

3 24 

2f^d? 4W I .^^r. .^^. 

' X ^=--Wfd, (66) 



24 ^d: 

The great objection to this form of pivot is the 
uneven wear due to the difference in velocity between 
center and circumference. 



MACHINE DESIGN. 7 1 

/It. Flat Collar. 

lyet d2= inside diameter 
Integrating as in equation (a) above, but using 

limits 



and 







— ^and — ^ we have 

2 2 






M= 


, dl-dl 
=2fp" 2 4 


this 


case 








P = - 


4W 




'(d^-d|) 




M= 


= ^Wf 


d^-d^ 

d?-d^ 



(67) 



42. Conical Pivot. 

Let a = angle of inclination to the vertical. 



M-p ' rA 




X 


\ \ I: 


\ 


^ — 


■v 


-^ 



dW= 



Fig. 27. 

4W 
-(d?-d^) 



As in the case of a 
flat ring the intensity 
of the vertical pres- 
sure is 

4W 



Kd?-dD 



and the vertical pres- 
sure on an elementary 
ring of the bearing 
surface is 



X 2 TT rdrizz 



8Wrdr 



dj~d 



As seen in Fig. 27 the normal pressure on the 
elementary ring is 

dP= -^ _ 8Wrdr 

sina (dj— d2)sina 



72 MACHINE DESIGN. 

The friction on the ring is fdP and the moment of 
this friction is 

,-- , ,_ SWf rMr 
dM=frdP^ ^^,_^.^^.^^ 

M= ,./^!. PrMr 



(df-d^)sinaJ d 



2 

sma dj— A\ 

TT 

As a approaches — the value of M approaches that of 

a flat ring, and as a approaches o the value of M ap- 
proaches 00 . 

If d2 =o we have 

Ayr T/ Wfd ,^ , 

M=^-^ (69) 

sina 

The conical pivot also wears unevenly, ucually 
assuming a concave shape as seen in profile. 

43. Schiele's Pivot. By experimenting with a 
pivot and bearing made of some friable material, it was 
shown that the outline tended to become curved as 
shown in Fig. 29, This led to a mathematical investi- 
gation which showed that the curve would be a trac- 
trix under certain conditions. 



This curve may be traced me- ,5 

chanically as shown in Fig. 28. I 

Let the weight W be free to j 

move on a plane. Let the string ; 

SW be kept taut and the end S • 

moved along the straight line SL. j 

Then will a pencil attached to the 1 

center of W trace on the plane a ^ L 
tractrix whose axis is SL. Fig- ^8. 



o 

w 



MACHINE DESIGN, 



73 







r, - 




— >1 


s 


*r_.~j/p 


/^ 


--w 




/ d 


w 


> 


"~d 


Q 








L 











In Fig. 29 let SW=length 
of string = r^ and let P be any 
point in the curve. Then it is 
evident that the tangent PQ 
to the curve is a constant and 

= 1*1 

r 



Also 



sin ^' 



Let a pivot be generated 
by revolving the curve around 
its axis SL. As in the case ol 
the conical pivot it can be 
^^S' 29. proved that the normal pres- 

sure on an element of convex surface is 



dP= 



8Wrdr 



(d?-d^)sin^ 



(a) 



Let the normal wear of the pivot be assumed to 
be proportional to this normal pressure and to the 
velocity of the rubbing surfaces, /. e, normal wear 
proportional to pr, then is the vertical wear 

proportional to '^ 



But- 



is a constant, there- 



sin<? * sin<? 

fore the vertical wear will be the same at all points. 
This is the characteristic feature and advantage of this 
form of pivot. 

As shown in equation (a) 

8Wr,dr 



dP: 



dM: 



8Wfr,rdr 



and 



M = 



8Wfr, rl-vl _ Wfd, 



d^-d 



(70) 



M is thus shown to be independent of d^ or of the 



length of pivot used. 



74 MACHINE DESIGN. 

This pivot is sometimes wrongly called anti- 
friction. As will be seen by comparing equations (66) 
and (70) the moment of friction is fifty per cent, greater 
than that of the common flat pivot. 

The distinct advantage of the Schiele pivot is in 
the fact that it maintains its shape as it wears and is 
self-adjusting. It is an expensive bearing to manu- 
facture and is seldom used on that account. 

It is not suitable for a bearing where most of the 
pressure is side ways. 

44. Multiple Bearings. To guard against abrasion 
in flat pivots a series of rubbing surfaces which divide 
the wear is sometimes provided, Several flat discs 
placed beneath the pivot and turning indifferently, 
may be used. Sometimes the discs are made alter- 
nately of a hard and a soft material. Bronze, steel and 
raw hide are the more common materials. 

Notice in this connection the button or washer at 
the outer end of the head spindle of an engine lathe 
and the loose collar on the main journal of a milling 
machine. Pivots are usually lubricated through a hole 
at the center of the bearing and it is desirable to have 
a pressure head on the oil to force it in. 

The compound thrust bearing generally used for 
propeller shafts consists of a number of collars of the 
same size forged on the shafts at regular intervals and 
dividing the end thrust between them, thus reducing 
the intensity of pressure to a safe limit without mak- 
ing the collars unreasonably large. 

A safe value for p the intensity of pressure is, ac- 
cording to Whitham, 60 lbs. per sq. in. for high speed 
engines. 

A table given by Prof. Jones in his book on 
Machine Design shows the practice at the Newport 
News ship-yards on marine engines of from 250 to 
5000 H. P. The outer diameter of collars is about one 
and one-half times the diameter of the shafts in each 
case and the number of collars used varies from 6 in 



MACHINE DESIGN. 75 

the smallest engine to ii in the largest. The pressure 
per sq. in. of bearing surface varies from 1 8 to 46 lbs. 
with an average value of about 32 lbs. 

EXAMPLES. 

1 . Design and draw to full size a Schiele pivot for 
a water wheel shaft 4 inches in diameter, the total 
length of the bearing being 3 inches. 

Calculate the horse-power expended in friction if 
the total vertical pressure on the pivot is two tons and 
the wheel makes 150 revs, per min. and assuming f = 
.25 for metal on wet vvood. 

2 . Design a compound thrust bearing for a pro- 
peller shaft the diameters being 14 and 21 inches, the 
total thrust being 80000 lbs. and the pressure 40 lbs. 
per sq. in. 

Calculate the horse-power consumed in friction 
and compare with that developed if a single collar of 
same area had been used. Assume f=.o5 and revs, 
per min. = 120. 



BALL AND ROLLER BEARINGS. 

45. General Principles. The object of interpos- 
ing a ball or roller between a journal and its bearing, 
is to substitute rolling for sliding friction and thus to 
reduce the resistance. This can be done only partially 
and by the observance of certain principles. In the 
first place it must be remembered that each ball can 
roll about but one axis at a time ; that axis must be 
determined and the points of contact located accord- 
ingly, 

Secondly, the pressure should be approximately 
normal to the surfaces at the points of contact. 

Finally it must be understood, that on account of 
the contact surfaces being so minute, a comparatively 
slight pressure will cause distortion of the balls and an 
entire change in the conditions. 

46. Journal Bearings. These may be either two, 
three or four point, so named from the number of 
points of contact of each ball. 

The axis of the ball may be assumed as parallel or 
inclined to the axis of the journal and the points of 
contact arranged accordingly. The simplest form con- 
sists of a plain cylindrical journal running in a bearing 
of the same shape and having rings of balls interposed. 
The successive rings of balls should be separated by 
thin loose collars to keep them in place. These collars 
are a source of rubbing friction, and to do away with 
them the balls are sometimes run in grooves either in 
journal, bearing or both. 

Fig. 30 shows a bearing of this type, there being 
three points of contact and the axis of ball being par- 



allel to that of journal 



MACHINE DESIGN. 



77 




The bearings so 
far mentioned have 
no means of ad- 
justment for wear, 
onical bearings, 
or those in which 
the axes of the 
^. balls meet in a 

^^S' 30. common point, 

supply this deficiency. In designing this class of bear- 
ings, either for side or end thrust, the inclination of 
the axis is assumed according to the obUquity desired 
and the points of contact are then so located that there 
shall be no slipping. 

Fig. 31 illustrates a common form of adjustable or 
cone bearing and shows the method of designing a 
three point contact. A C is the axis of the cone, while 
the shaded area is a section of the cup, so called. Let 
a and b be two points of contact between ball and cup. 
Draw the line a b and produce to cut axis in A . Through 
the center of ball draw the line A B ; then will this be 
the axis of rotation of the ball and a c, b d will be the 
projections of two circles of rotation. As the radii of 
theae circles have the same ratio as the radii of revo- 
lution an, b m , there will be no slipping and the ball 



will roll as a cone inside another cone. The exact 
cation of the third point of contact is not material. 
it were at c, too much 
pressure would come on 
the cup at b ; if at d 
there would be an ex- 
cess of pressure at a but 
the rolling would be cor- 
rect in either case. A 
convenient method is to 
locate p by drawing A D 
tangent to ball circle as 
shown. It is recom- 
mended however that 
the two opposing sur- 



lo- 
If 




t--€^ -^ 



3, 




78 MACHINE DESIGN. 

faces at p and b or a should make with each other an 
angle of not less than 25"^ to avoid sticking of the ball. 

To convert the bearing just shown to four point 
contact, it would only be necessary to change the one 
cone into two cones tangent to the ball at c and d. 

To reduce it to two point contact the points a and b 
are brought together to a point opposite p. As in this 
last case the ball would not be confined to a definite 
path it is customary to make one or both surfaces 
concave conoids with a radius about three fourths the 
diameter of the ball. See Fig. 32. 

47. Step -Bearings. 

The same principles ap- 
ply as in the preceding 
article and the axis and 
points of contact may 
be varied in the same^- 
way. The most com- m 
mon form of step- i 
bearing consists of two 
flat circular plates sep- 
arated by one or more 
rings of balls. Each 
ring must be kept in 
place by one or more 

loose retaining collars, and these in turn are the cause 
of some sHding friction. This is a bearing with two 
point contact and the balls turning on horizontal axes. 
If the space between the plates is filled with loose 
balls, as is sometimes done, the rubbing of the balls 
against each other will cause considerable friction. 

To guide the balls without rubbing friction three 
point contact is generally used. 

Fig. 33 illustrates a bearing of this character. 
The method of design is shown in the figure the prin- 
ciple being the same as in Fig. 31. By comparing the 
lettering of the two figures the similarity will be read- 
ily seen. 




MACHINE DESIGN 



79 




This last bearing may be converted to four point 
contact by making the upper collar of the same shape 
as the lower. To guide the balls in two point contact 
use is sometimes made of a cage ring, a flat collar 
drilled with holes just a trifle larger than the balls and 

disposing them 
either in spirals 
or in irregular 
order. 

This method has 
the advantage of 
making each ball 
move in a path of 
different radius 
thus securing more 
even wear for the 
plates. 

48. Materials and Wear. The balls themselves 
are always made of steel, hardened in oil, tempered and 
ground. They are usually accurate to within one ten 
thousandth of an inch. The plates, rings and journals 
must be hardened and ground in the same way and 
perhaps are more likely to wear out or fail than the 
balls. A long series of experiments made at the Case 
School of Applied Science on the friction and endur- 
ance of ball step-bearings showed some interesting pe- 
culiarities. 

Using flat plates with one circle of quarter inch 
balls it was found that the balls pressed outward on 
the retaining ring with such force as to cut and in- 
dent it seriously. This was probably due to the fact 
that the pressure slightly distorted the balls and 
changed each sphere into a partial cylinder at the 
touching points. While of this shape it would tend to 
roll in a straight line or a tangent to the circle. 
Grinding the plates slightly convex at an angle of one 
to one and-a-half degrees obviated the difficulty to a 
certain extent. Under even moderately heavy loads 



8o MACHINE DESIGN. 

the continued rolling of the ring of balls in one path 
soon damaged the plates to such an extent as to ruin 
the bearing. 

A flat bearing filled with loose balls developed 
three or four times the friction of the single ring and 
a three point beaKing similar to that in Fig. 33 showed 
more than twice the friction of the two point. 

A flat ring cage such as has already been described 
was the most satisfactory as regards friction and en- 
durance. 

The general conclusions derived from the experi- 
ments were that under comparatively light pressures 
the balls are distorted sufficiently to seriously disturb 
the manner of rolling and that it is the elasticity and 
not the compressive strength of the balls which must 
be considered in designing bearings. 

49. Design of Bearings. Figures on the direct 
crushing strength of steel balls have little value for the 
designer. For instance it has been proved by numer- 
ous tests that the average crushing strengths of }^ inch 
and ^ inch balls are about 7500 lbs. and 15000 lbs. 
respectively. Experiments made by the writer show 
that a ^ inch ball loses all value as a transmission 
element on account of distortion, at any load of more 
than 100 lbs. 

Prof. Gray states, as a conclusion from some ex- 
periments made by him, that not more than 40 lbs. per 
ball should be allowed for Y^ inch balls. 

This distortion doubtless accounts for the failure 
of theoretically correct bearings to behave as was ex- 
pected of them. Ball bearings should be designed as 
has been explained in the preceding articles and then 
only used for light loads. 

50. Roller Bearings. The principal disadvantage 
of ball bearings lies in the fact that contact is only at 
a point and that even moderate pressure causes exces- 
sive distortion and wear. The substitution of cylin- 
ders or cones for the balls is intended to overcome this 
difficulty. 



MACHINE DESIGN. 8 1 

The simplest form of roller bearing consists of a 
plain cylindrical journal and bearing with small cylin- 
drical rollers interposed instead of balls. There are two 
difl&culties here to be overcome. The rollers tend to 
work endways and rub or score whatever retains them. 
They also tend to twist around and become unevenly 
worn or even bent and broken, unless held in place by 
some sort of cage. In short they will not work prop- 
erly unless guided and any form of guide entails sliding 
friction. The cage generally used is a cylindrical 
sleeve having longitudinal slots which hold the rollers 
loosely and prevent their getting out of place either 
sideways or endways. 

The use of balls between the rollers at the ends 
has been tried with some degree of success. Large rol- 
lers have been turned smaller at the ends and the 
bearings then formed allowed to turn in holes bored 
in revolving collars. These collars must be so fastened 
or geared together as to turn in unison. 

SI. Hyatt Rollers. The tendency of the rollers to 
get out of alignment has been already noticed. The 
Hyatt roller is intended by its flexibility to secure 
uniform pressure and wear under such conditions. It 
consists of a flat strip of steel wound spirally about 
a mandrel so as to form a continuous hollow cylinder. 
It is true in form and comparatively rigid against com- 
pression, but possesses sufficient flexibility to adapt 
itself to slight changes of bearing surface. This bear- 
ing is readily caged by running rods through the rol- 
lers and riveting them to collars at the ends. 

Experiments made by the Franklin Institute show 
that the Hyatt roller possesses a great advantage in 
eflSciency over the solid roller. 

Testing ^ inch rollers between flat plates under 
loads increasing to 550 lbs. per linear inch of roller 
developed co-efficients of friction for the Hyatt roller 
from 23 to 51 per cent, less than for the solid roller. 



82 MACHINE DESIGN. 

Subsequent examination of the plates showed also a 
much more even distribution of pressure for the for- 
mer. 

52. Roller Step Bearings. In article 48 attention 
was called to the fact that the balls in a step-bearing 
under moderately heavy pressures tend to become 
cylinders or cones and to roll accordingly. This has 
suggested the use of small cones in place of the balls, 
rolling between plates one or both of which is also 
conical. A successful bearing of this kind with short 
cylinders in place of cones is used by the Sprague- 
Pratt Elevator Co., and is described in the American 
Machinist for June 27, 1901. The rollers are arranged 
in two spiral rows so as to distribute the wear evenly 
over the plates and are held loosely in a flat ring cage. 
This bearing has run well in practice under loads 
double those allowable for ball bearings, or over 100 
lbs. per roll for rolls one-half inch in diameter and 
one-quarter inch long. 



SHAFTING, COUPLINGS AND HANGERS. 

53. Strength of Shafting. 

I^et D = diameter of the driving pulley or gear. 
N= number revs, per minute. 
P= force applied at rim. 
T=^ twisting moment. 

The distance through which P acts in one minute 
is ttDN inches and work— Pt^DN in. lbs. per min. 

PD 

But = T the moment, and 27rN= the angular 

2 

velocity. 

. • . work = moment X angular velocity 
One horse power = 33000 ft. lbs. per min. 
= 396000 in. lbs. per min. 

HP= ^^^^ _ 27rTN 

396000 396000 
TN 

^^=^5 ^''' 

also T=^3-^ (^^^ 

126050 HP 

DN ^'■^^ 

The general formula for a circular shaft exposed 
to torsion alone is 

where N = no. revs, per min. 



84 



MACHINE DESIGN, 



Substituting in formula for d 

d = ^ I321000HP 



SN 



nearly (74) 



It 
ing as 



S may be given the following values : 
45000 for common turned shafting. 
50000 for rolled iron or soft steel. (0.15 C) 
65000 for machinery steel. (0.55 C) 

is customary to use factors of safety for shaft- 
follows: f 
Headshafts or prime movers 15 
lyine shafting 10 
Short counters 6 
The large factor of safety for head shafts is used 
not only on account of the severe service to which 
such shafts are exposed, but also on account of the in- 
convenience and expense attendant on failure of so 
important a part of the machinery. The factor of 
safety for line shafting is supposed to be large enough 
to allow for the transverse stresses produced by weight 
of pulleys, pull of belts, etc., since it is impracticable 
to calculate these accurately in most cases. 

Substituting the valuoe of S and introducing fac- 
tors of safety, we have the following formulas for the 
safe diameters of the various kinds of shafts. 

TABLE XV. 



Kind of 


Material. 


Shaft. 


Common Iron 


Soft Steel 


Machy. Steel 


Head Shaft. 

Line Shaft. 

Counter 

Shaft. 


, (HP 
3 HP 
3 fHP 


, HP 

3 HP 


3 fHP 

^ » HP 

3 HP 



MACHINE DESIGN. 85 

In case there is a known bending moment M, com- 
bined with a known twisting moment T, then a re- 
sultant twisting moment 

T=M-f x/m^ + T^ 

is to be substituted for T in the above formulas. 



54. Couplings. The flange or plate coupling is 
most commonly used for fastening together adjacent 
lengths of shafting. 



Fig. 34 shows the proportions of such a coupling. 
The flanges are turned accurately on all sides, are 
keyed to the shafts and the two are centered by the 
projection of the shaft from one part into the other as 

shown at A. The 
bolts are turned 
to fit the holes 
loosely so as not 
to interfere with 
the alignment. 

The projecting 
rim as at B pre- 
vents danger 
from belts catch- 
ing on the heads 
and nuts of the 
bolts. 




^sssij 



Fig- 34" 



The faces of this coupling should be trued up 
in a lathe after being keyed to the shaft. 

The sleeve coupling is less clumsy than the fore- 
going but is rather more complicated and expensive. 



86 



MACHINE DESIGN. 



In Fig. 35 is illustrated a neat and efifective coup- 
lino^ of this type. It consists of the sleeve S bored 
with two tapers and two threaded ends as shown. 
The two conical, split bushings B B are prevented from 
turning by the feather key K and are forced into the 
conical recesses by the two threaded collars C C and 
thereby clamped firmly to the shaft. The key K also 
nicks slightly the center of the main sleeve S, thus 
locking the whole combination. 




Fig. 35. 

Couplings similar to this have been in use in the 
Jnion Steel Screw Works, Cleveland, Ohio, for many 
years and have given good satisfaction. 

In another form of sleeve coupling the sleeve is 
split and clamped to the shaft by bolts passing through 
the two halves. 

55. Coupling Bolts. The bolts used in the ordi- 
nary flange couplings are exposed to shearing, and 
their combined shearing moment should equal the 
twisting moment on the shaft. 

Let n=number of bolts. 
di=diameter of bolt. 
D= diameter ot bolt circle. 
We will assume that the bolt has the same shear- 
ing strength as the shaft. The combined shearing 
strength of the bolt is .7854dinS and their moment of 
resistance to shearing is 

.7854d^nSx — =.3927Dd,'nS 



MACHINK DESIGN. 87 

This last should equal the torsion moment of the 

shaft or .3927Dd?nS= 

5.1 

Solving for d^ and assuming D = 3d as an average 

value we have d^= — — •••(75) 

V 6n 

In practice rather larger values are used than 

would be given by the formula. 

56. Shafting Keys. The moment of the shearing 
stress on a key must also equal the twisting moment 
of the shaft. 

Let b= breadth of a key. 
i=length of key. 
h= total depth of key. 
S'= shearing strength of key. 

The moment of shearing stress on key is 

bis'xA=_Mis: 

2 2 

Q 1 3 J 

and this must equal — Usually b= — 

5-1 4 

For shafts of machine steel S=S', and for iron 
shafts S=^S' nearly as keys should always be of 
steel. 

Substituting these values and reducing : 

For iron shafting l=i. 2d nearly. 

For steel shafting l=i.6d nearly, as the least 
lengths of key to prevent its fail- 
ing by shear. 

If the key way is to be de- 
signed for uniform strength, the 
shearing area of the shaft on the 
line AB Fig. 35a should equal 
the shearing area of the key, if 
shaft and key are of the same 
material and AB=CD=b. 




S8 MACHINE DESIGN. 

These proportions will make the depth of key way 
in shaft about = ^b and would be appropriate for a 
square key. 

To avoid sucii a depth of key way which might 
weaken the shaft, it is better to use keys longer than 
required by preceding formulas. In American practice 
the total depth of key rarely exceeds ^b and one-half 
of this depth is in shaft. 

To prevent crushing of the key the moment of 

the compressive strength of half the depth of key must 

equal T. 

d Ih ^ Sd^ 
or — X - — X Sc= (a) 

2 2 ' 5.1 

where Sc is the compressive strength of the key. 

For iron shafts Sc =28 

and for steel shafts Sc =— S 

2 

Substituting values of S© and assuming h=^b 
= J 2 d we have 

Iron shafts l=2.5d nearly 

Steel shafts 1= 3 j^d nearly, as the least 

length for flat keys to prevent lateral crushing. 

57. Hangers and Boxes. As shafting is usually 
hung to the ceiling and walls of buildings it is neces- 
sary to provide means for adjusting and aligning the 
bearings as the movement of the building disturbs 
them. Furthermore as line shafting is continuous and 
is not perfectly true and straight, the bearings should 
be to a certain extent self-adjusting. Reliable experi- 
ments have shown that usually one-half of the power 
developed by an engine is lost in the friction of shafting 
and belts. It is important that this loss be prevented 
as far as possible. 

The boxes are in two parts and may be of bored 
cast iron or lined with Babbitt metal. They are 
usually about four diameters of the shaft in length and 
are oiled by means of a well and rings or wicks. 



MACHINE DESIGN, 



89 



(See Chapter 7.) The best method of supporting the 
box in the hanger is by the ball and socket joint ; all 
other contrivances such as set screws are but poor 
substitutes. 



Fig. 36 shows the 
usual arrangement 
of the ball and 
socket. 

A and B are the 
two parts of the 
box. The center 
is cast in the shape 
of a partial sphere 
with C as a center 
as shown by the 
dotted lines. The 
two sockets S S 
can be adjusted 
vertically in the hanger by means of screws and lock 
nuts. The horizontal adjustment of the hanger is 
usually effected by moving it bodily on the support, 
the bolt holes being slotted for this purpose. 




Fig. 36. 



Counter shafts are short and light and are not 
subject to much bending. Consequently there is not 
the same need of ad- 
justment as in line 
shafting. 

In Fig, 37 is il- 
lustrated a simple 
bearing for counters. 
The solid cast iron 
box B with a spherical 
center is fitted directly 
in a socket in the 
hanger H and held in 
position by the cap C 
and a set screw. Fig. 37. 




90 MACHINE DESIGNc 

There is not space here to show the various forms 
of hangers and floor stands and reference is made to 
the catalogues of manufacturers. Hangers should be 
symmetrical, i. e. the center of the box should be in a 
vertical line with center of base. They should have 
relatively broad bases and should have the metal dis- 
posed to secure the greatest rigidity possible. Cored 
sections are to be preferred. 

EXAMPLES. 

1. Calculate the safe diameters of head shaft and 
three line shafts for a factory, the material to be rolled 
iron and the speeds and horse powers as follows : 
Head shaft loo HP 200 revs, per min. 
Machine shop 30 HP 120 revs, per min. 
Pattern shop 50 HP 250 revs, per min. 
Forge shop 20 HP 200 revs, per min. 

2. Determine the horse power of at least two lines 
of shafting whose speed and diameter are known. 

3. Design and sketch to scale a flange coupling 
for a three inch line shaft including bolts and keys. 

4. Design a sleeve coupling for the foregoing, 
different in principle from the one shown in the text. 

5. Select the line shaft hanger which you prefer 
among those in the laboratories and make sketch and 
description of the same. 

6. Do. for a countershaft hanger. 

7. Explain in what way a floor stand differs from 
a hanger. 



GEARS, PULLEYS AND FLY WHEELS. 

58. Gear Teeth. The teeth of gears are made in 
three ways and are accordingly known as pattern 
molded, machine molded and machine cut, the first 
being the least accurate in form and the last the most 
so. 



Let circular pitch 
diameter pitch 

pitch number 

addendum 

flank 

clearance 

height 

width 

space 



P 
P-^d 

TT 

TT _ I 

a 

f 

f— a==c 

f+a==h 

w 

p— w=s 



The following table gives the ordinary proportions 
in use for the three kinds of teeth. 

TABLE XVL — PROPORTIONS OF GEAR TEETH. 



Kind of 


Adden- 
dum. 

a 


Flank. 


Clearance 


Height . 


Width. 


*paoe. 


Tooth. 


f 


C 


h 


W 


S 


Pattern 
Molded 


•32 P 


•38 P 


.06 p 


•7P 


•47 P 


•53 P 


Machine 
Molded 


•32 P 


•38 P 


.06 p 


•7P 


.48 p 


•52 p 


Machine 
Cut 


d 


ij^d 


m 


2Hd 


•5P 


•5P 



92 



MACHINE DESIGN. 



Ill making machine cut teeth some mechanics 
prefer to make the clearance = 2V p which is slightly 
more than }id. 

The diameter pitch bears the same relation to the 
diameter as the circular pitch does to the circumfer- 
ence, and may be obtained by dividing the diameter 
by the number of teeth. 



Its value is evidently d= 



•3183 p. 



The pitch number is simply the reciprocal of d 
and shows the number of teeth in the wheel per inch 
of diameter. 

In modern practice the pitch number is either a 
whole number or a common fraction. 

The following table gives values of the diameter 
and circular pitches for all the pitch numbers in use. 

TABLE XVII.— PITCH OF GEAR TEETH. 



Pitch 


Diam - 


Circu- 


Pitch 


Diam- 


Circu - 


Pitch 


Diam- 


Circu - 


Num- 


eter 


lar 


Num- 


eter 


lar 


Num- 


eter 


lar 


ber. 


Pitch 


Pitch 


ber. 


Pitch 


Pitch 


ber. 


Pitch 


Pitch 


I 


d 


P 


I 


d 


P 


I 


d 


P 


d 




1 


d" 






d 






K 


2. 


6.2832 


7 


.1429 


.4488 


24 


.0417 


.1309 


u 


1-3333 


4.1888 


8 


-125 


•3927 


26 


.0385 


.1208 


1 


I. 


3.1416 


9 


.1111 


•3491 


28 


.0357 


.1122 


IX 


.8 


2.5133 


10 


.1 


.3142 


30 


.0333 


.1047 


^'A 


.6667 


2.0944 


II 


.0909 


.2856 


32 


.0312 


.0982 


i^ 


•5714 


1.7952 


12 


-0833 


.2618 


34 


.0294 


.0924 


2 


.5 


1.5708 


13 


.0769 


.2417 


36 


.0278 


.0873 


2H 


.4444 


1-3963 


14 


.0714 


.2244 


38 


.0263 


.0827 


2^ 


.4 


1.2566 


15 


.0667 


.2094 


40 


.025 


.0785 


2H 


•3366 


I. 1424 


16 


.0625 


.1963 


42 


.0238 


.0748 


3 


•3333 


1.0472 


17 


.0588 


.1848 


44 


.0227 


.0714 


3>^ 


.2857 


.8976 


18 


.0555 


-1745 


46 


.0217 


.0683 


4 


.25 


.7854 


19 


.0526 


.1653 


48 


.0208 


.0654 


5 


.2 


.6283 


20 


.05 


.1571 


50 


.02 


.0628 


6 


.1667 


.5236 


22 


.0455 


.1428 


56 


.0179 


.0561 



MACHINE DESIGN. 



93 



The proportions given in Table XVI. are those 
most usual in practice, but many good authorities 
recommend a shorter tooth as giving less obHquity and 
sliding friction and as being much stronger. 

The length generally recommend-^d in such cases is 
equal to one-half the circular pitch plus the clearance. 
Short teeth are being more used every year. 



Strength of Teeth. 

P = total driving pressure on wheel at pitch 



59. 

Let 

circle. This may be distributed over two or more 
teeth, but the chances are against an even distribution. 

Again, in designing a set of gears the contact is 
likely to be confined to one pair of teeth in the smaller 
pinions. 

Each tooth should therefore be made strong 
enough to sustain the whole pressure. 

Rough Teeth, The teeth of pattern molded gears 
are apt to be more or less irregular in shape, and are 
especially liable to be thicker at one end on account of 
the draft of the pattern. 

In this case the entire pressure may come on the 
outer corner of a tooth and tend to cause a diagonal 
fracture. 

Let C in Fig. 39 be the point of application of the 
pressure P, and AB the line of probable fracture. 

B 




94 



MACHINE DESIGN 



Use the notation of Fig. 38 and the proportions 
for pattern molded teeth in Table XVI. 

The bending moment at 
section AB is M=Py, and 
the moment of resistance 

is M' = -i-Sxw' 

where S = safe transverse 
strength of material. 

Py =4-Sxw^ 
6 




and 



Fig. 38. 



S = 



6Py 



(a) 



when 



If P and w are constant, then S is a maximum 

y 



IS a maximum. 



But 



y = h sin a and x=- 



cosa 

y 

— = sin« cosa which is a 



maximum when «= 45"* and — = J^ 

X ^ 

^P 

Substituting this value in (a) we have S = -^ 

7 p 
But in this case w=.47p and therefore S = -^ — i 

^'^ .22Ip* 

and p=3.684 f— 

(~p 
Diameter pitch, d= 1.173 — 

S^ 

P 

Unless machine molded teeth are very carefully- 
made, it may be necessary to apply this rule to them 
as well. 



Pitch number 



' "^ = -853 J 



(76) 
(77) 
(78) 



MACHINK DESIGN. 95 

Cut Gears, With careful workmanstiip machine 
molded and machine cut teeth should touch along the 
whole breadth. In such cases we may assume a line of 
contact at crest of tooth and a maximum bending mo- 
ment 

M==Ph 

The moment of resistance at base of tooth is 
M^ = >^Sbw^ 
when b is ^he breadth of tooth. 

In most teeth the thickness at base is greater than 
w, but in radial teeth it is less. Assuming standard 
proportions for cut gears : 

h — 2}id =.6765p 
W--.5P 
and substituting above : 

A A T. Sbp^ 
.6765 Pp=-^ 

P = .o6i6bSp .(79) 

The above formula is general whatever the ratio 
of breath to pitch. The general practice in this country 
is to make b=3p 

Substituting this value of b in (79) and reducing: 
p=2.326 I— (8o> 

or about two thirds the value obtained in Case I. 

|P" 
Diameter pitch d=.74 I— - ,. (81) 

I /"s~ 
Pitch number ~d'=^-35j-^ (82) 

60. Lewis' Formulas. The foregoing formulas 
can only be regarded as approximate, since the strength 
of gear teeth depends upon the number of teeth in the 
wheel; the teeth of a rack are broader at the base 
and consequently stronger than those of a pinion. 
This is more particularly true of epicycloidal teeth. 



96 MACHINE DESIGN. 

Mr. Wilfred Lewis has deduced formulas which take into 
account this variation. For cut spur gears of standard 
dimensions the Lewis formula is as follows : 

P=bSp (.124-^^51) (83) 

n 

where n = number of teeth. 

This formula reduces to the same as (79) for 
n=i4 nearly. 

Formula (79) would then properly apply only to 
small pinions, but as it would err on the safe side for 
(arger wheels, it can be used where great accuracy is 
not needed. The same criticism applies to formulas 
(80) (81) and (82). 

The value of S used should depend on the ma- 
terial and on the speed. 

The following values are recommended for cast 
iron and cast steel. 

Ivinear Velocity o 500 1000 1500 2000 

ft. per minute 

Cast Iron 6000 4500 2500 2000 1800 
Cast Steel 15000 loooo 7000 5000 4500 

Good bronze will have about the same strength 
as the steel. The smaller values of S at the higher 
speeds are to allow for the blows and shocks which 
always occur in quick running gears. 

61 Experimental Data. Inth^ American Mac hinisf 
for Jan. 14, 1897 ^^^ given the actual breaking loads 
of gear teeth which failed in service. The teeth had an 
average pitch of about 5 inches a breadth of about 18 
inches and the rather unusual velocity of over 2000 ft. 
per minute. The average breaking load was about 
15000 lbs. there being an average of about 50 teeth on 
the pinions. Substituting these values (83) and solv- 
ing we get 

8=1575 lbs. 



MACHINE DESIGN. 97 

This very low value is to be attributed to the con- 
dition of pressure on one corner noted in Art. 59. 
Substituting in formula for such a case. 

S=7^^7^ -8150 

This all goes to show that it is well to allow large 
factors of safety for rough gears, especially when the 
speed is high. 

Experiments have been made ou the static strength 
of rough cast iron gear teeth at the Case School of 
AppUed Science by breaking them in a testing machine. 
The teeth were cast singly from patterns, were two 
pitch and about 6 inches broad. The patterns were 
constructed accurately from templates representing 
15^ involute teeth and cycloidal teeth constructed with 
describing circle one-half the pitch circle of 15 teeth; 
the proportions used were those given for standard cut 
gears. 

There were in all 41 cycloidal teeth of shapes cor- 
responding to wheels of 15-24-36-48-72-120 teeth 
and a rack. There were 28 involute teeth correspond- 
ing to numbers above given omitting the pinion of 15 
teeth. 

The pressure was applied by a steel plunger tan- 
gent to the surface of tooth and so pivoted as to bear 
evenly across the whole breadth. The teeth were in- 
clined at various angles so as to vary the obliquity 
from o to 25° for the cycloidal and from 15° to 25^ for 
the involute. The point of application changing accord- 
ingly from the pitch line to the crest of the tooth. 
From these experiments the following conclusions were 
drawn; 

1 . The plane of fracture is approximately parallel 
to line of pressure and not necessarily at right angles 
to radial line, through center of tooth. 

2. Corner breaks are likely to occur even when 
the pressure is apparently uniform along the tooth. 
There were fourteen such breaks in all. 



98 MACHINE DESIGN. 

3. With teeth of dimensions given, the breaking 
pressure per tooth varies from 25000 lbs. to 50000 lbs. 
for cycloids as the number of teeth increases from 15 
to infinity; the breaking pressure for involutes of the 
same pitch varies from 34000 lbs to 80000 lbs. as the 
tooth number increases from 24 to infinity. 

4. With teeth as above the average breaking pres- 
sure varies from 50000 lbs. to 26000 lbs. in the cycloids 
as the angle changes from 0° to 25^ and the tangent 
point moves from pitch line to crest, with involute 
teeth the range is between 64000 and 39000 lbs. 

5. Reasoning from the figures just given, rack 
teeth are about twice as strong as pinion teeth and in- 
volute teeth have an advantage in strength over cy- 
cloidal of from forty to fifty per cent. The advantage 
of short teeth in point of strength can also be seen. 
The modulus of rupture of the material used was about 
36000 lbs. Values of S calculated from Lewis' formula 
for the various tooth numbers are quite uniform and 
average about 40000 lbs. forcycloidal teeth. Involute 
teeth are to-day generally preferred by manufacturers. 
William Sellers & Co. use an obliquity of 20^ instead 
of 14^ or 15^ the usual angle. 

62. Teeth of Bevel Gears. There have been many 
formulas and diagrams proposed for determining the 
strength of bevel gear teeth, some of them being very 
complicated and inconvenient. It will usually answer 
every purpose from a practical standpoint, if we treat 
the section at the middle of the breadth of such a tooth 
as a spur wheel tooth and design it by the foregoing 
formulas. The breadth of the teeth of a bevel gear 
should be about one-third of the distance from the base 
of the cone to the apex. 

One point needs to be noted; the teeth of bevel 
gears are stronger than those of spur gears of the 
same pitch and number of teeth since they are de- 
veloped from a pitch circle having an element of the 
normal cone as a radius. To illustrate we will suppose 
that we are designing the teeth of a miter gear and 



MACHINE DESIGN. 99 

that the number of teeth is 32. In such a gear the 
element of normal cone is \/~2~times the radius. The 
actual shape of the teeth will then correspond to those 
of a spur gear having 32^/T"== 45 teeth nearly. 

Note.— In designing the teeth of gears where the 
number is unknown, the approximate dimensions may- 
first be obtained by formula (80) and then these values 
corrected by using Lewis' formula. 

EXAMPLES. 

1 . The drum of a hoist is 8 ins. in diameter and 
makes 5 revs, per minute. The diameter of gear on 
the drum is 36 inches and of its pinion 6 ins. The 
gear on the counter shaft is 24 ins. in diameter and its 
pinion is 6 ins. in diameter. The gears are all rough. 

Calculate the pitch and number of teeth of each 
gear, assuming a load of one ton on drum chain and 
S = 6000, Also determine the horse-power of the 
machine. 

2. Calculate the pitch and number of teeth of 
a cut cast steel gear 10 ins. in diameter, running at 
250 revs, per min. and transmitting 20 HP. 

3. A cast iron gear wheel is 30 ft. 6^ ins. in 
pitch diameter and has 192 teeth, which are machine 
cut and 30 ins. broad. 

Determine the circular and diameter pitches of the 
teeth and the horse-power which the gear will trans- 
mit safety when making 12 revs, per min. 

4. A two pitch cycloidal tooth, 6 ins. broad, 72 
teeth to the wheel, failed under a load of 38000 lbs. 
Find value of S by Lewis' formula. 

5. A vertical water wheel shaft is connected to 
horizontal head shaft by cast iron gears and transmits 
150 H P. The water wheel makes 200 revs, per min. 
and the head shaft 100. 

Determine the dimensions of the gears and teeth 
if the latter are approximately two pitch. 

LofC. 



lOO MACHINE DESIGN. 

63. Rim and Arms. The rim of a gear, especially 
if the teeth are cast, should have nearly the same 
thickness as the base of tooth, to avoid cooling strains. 

It is difficult to calculate exactly the stresses on 
the arms of the gear, since we know so little of the 
initial stress present, due to cooling and contraction. 
A hub of unusual weight is liable to contract in cooling 
after the arms have become rigid and cause severe 
tension or even fracture at the junction of arm and 
hub. 

A heavy rim on the contrary may compress the 
arms so as actually to spring them out of shape. Of 
course both of these errors should be avoided, and the 
pattern be so designed that cooling shall be simul- 
taneous in all parts of the casting. 

The arms of spur gears are usually made straight 
without curves or taper, and of a flat, elliptical cross- 
section, which offers little resistance to the air. To 
support the wide rims of bevel gears and to facilitate 
drawing the pattern from the sand, the arms are some- 
times of a rectangular or T section, having the greatest 
depth in the direction of the axis of the gear. For 
pulleys which are to run at a high speed it is import- 
ant that there should be no ribs or projections on arms 
or rim which will offer resistance to the air. Experi- 
ments by the writer have shown this resistance to be 
serious at speeds frequently used in practice. 

A series of experiments conducted by the author 
are reported in the Anierican Machinist for Sep. 22, 
1898, to which paper reference is here made. 

Twenty- four pulleys having 3^ inches face and 
diameters of 16, 20 and 24 inches were broken in a 
testing machine by the pull of a steel belt, the ratio of 
the belt tension being adjusted by levers so as to be 
two to one. Twelve of the pulleys were of the ordi- 
nary cast iron type having each six arms tapering and 
of an elliptic section. The other twelve were Medart 
pulleys with steel rims riveted to arms and having 
some six and some eight arms. Test pieces cast from 



MACHINE DESIGN. lOI 

the same iron as the pulleys showed an average modu- 
lus of rupture of 35800 for the cast iron and 50800 for 
the Medart. 

In every case the arm or the two arms nearest the 
side of the belt having the greatest tension, broke first 
showing that the torque was not evenly distributed by 
the rim. Measurements of the deflection of the arms 
showed it to be from two to six times as great on this 
side as on the other. The buckling and springing of 
the rim was very noticeable especially in the Medart 
pulleys. 

The arms of all the pulleys broke at the hub show- 
ing the greatest bending moment there as the strength 
of the arms at the hub was about double that at the 
rim. On the other hand some of the cast iron arms 
broke simultaneously at hub and rim showing a nega- 
tive bending moment at the rim about one-halt that at 
the hub. 

The following general conclusions are justified by 
these experiments : 

(a) The bending moments on pulley arms are not 
evenly distributed by the rim, but are greatest next 
the tight side of belt. 

(b) There are bending moments at both ends of 
arm, that at the hub being much the greater, the ratio 
depending on the relative stiffness of rim and arms. 

The following rules may be adopted for designing 
the arms of cast iron pulleys and gears : 

1. Multiply the net turning pressure, whether 
caused by belt or tooth, by a suitable factor of safety 
and by the length of the arm in inches. Divide this 
product by one-half the number of arms and use the 
quotient for a bending moment. Design the hub end 
of arm to resist this moment. 

2. Make the rim ends of arms one-half as strong 
as the hub ends. 



I02 MACHINE DESIGN. 

64. Sate speed for Wheels The centrifugal force 
developed in a rapidly revolving pulley or gear pro- 
duces a certain tension on the rim, and also a bending 
of the rim between the arms. We will first investigate 
the case of a pulley having a rim of uniform cross 
section. 

It is safe to assume that the rim should be capable 
of bearing its own centrifugal tension without assist- 
ance from the arms. 

Let D= mean diameter of pulley rim. 
t=thickness of rim. 
b=breadth of rim. 
w=weight of material per cu. in. 

= .26 lbs. for cast iron. 

= .28 lbs. for wrought iron or steel. 
n=number of arms. 
N= number revs, per min. 
v=r velocity of rim in ft. per sec. 

First let us consider the centrifugal tension alone. 
The centrifugal pressure per square inch of concave 
surface is 

Wv^ 

P = "^ — ^") 

where W is the weight of rim per square inch of con- 

D 
cave surface = wt, and r= radius in feet = ~^T~ 

The centrifugal tension produced in the rim hj 
this force is by formula (13) 

^— 2t 

Substituting the values of p, W and r and reducing: 

I2WV^ 






and v=^^- (85) 



MACHINE DESIGN. IO3 

For an average value of w=.27, (86) reduces to 



v^ 
S = — nearly. 

a convenient form to remember. 

If v/e assume S as the ultimate tensile strength, 
16500 lbs. for cast iron in large castings and 60000 lbs. 
for soft steel, then the bursting speed of rim is ; 

for a cast iron wheel 

v=4o6 ft. per sec (86) 

and for steel rim ^=775 ft. per sec (87) 

and these values may be used in roughly calculating 
the safe speed of pulleys. 

It has been shown by Mr. James B. Stanwood, 
in a paper read before the American Society of Me- 
chanical Engineers,^ that each section of the rim be- 
tween the arms is moreover in the condition of a beam 
fixed at the ends and uniformly loaded. 

This condition will produce an additional tension 
on the outside of rim. The formula for such a beam 
when of rectangular cross-section is 

Wl Sbd^ 

17=-^ (^) 

W in this case is the centrifugal force of the frac- 
tion of rim included between two arms. 

TrDbtW 

The weight of this fraction is ~ and its cen- 

TrDbtw 24V^ 247rbtwv^ 

trifugal force W = - X -^^orW=— ^^^^ 

ttD 

Also 1 — ~~z~~ and d=t 



* See Trans. A. S. M. B. Vol. XIV. 



I04 MACHINE DESIGN. 

Substituting these values in (b) and solving 
for S : 

Dwv" 
8=3.678-^^ (c) 

If w is given an average value of .27 then 

Dv' 
S = -^^ nearly (d) 

and the total value of the tensile stress on outer sur- 
face of rim is 

Dv' v' 
S'=^^ + j^nearly (88) 



Solving for v : 

V =.U . _i_ (89) 

X tn2^ 10 

In a pulley with a thin rim and small number of 
arms, the stress due to this bending is seen to be con- 
siderable. 

It must however be remembered that the stretch- 
ing of the arms' due to their own centrifugal force and 
that of the rim will to some extent diminish this bend- 
ing. Mr. Stanwood recommends a deduction of one- 
half from the value of S in (d) on this account. 

Prof, Gaetano Lanza has published quite an elab- 
orate mathematical discussion of this subject. (See Vol. 
XVI. Trans. Am. Soc. Mech. Engineers.) He shows 
that in ordinary cases the stretch of the arms will re- 
lieve more than one half of the stress due to bending, 
perhaps three-quarters. 



MACHINE DESIGN. I05 

65 Experiment^ on Fly Wheels. In order to de- 
termine experimentally the centrifugal tension and 
bending in rapidly revolving rims, a large number of 
small fly wheels have been tested to destruction at the 
Case School laboratories. In all ten wheels, fifteen 
inches in dameter and twenty-three wheels two feet in 
diameter have been so tested. An account of some of 
these experiments may be found in Trans. Am. Soc. 
Mech. Eng. Vol. XX. The wheels were all of cast iron 
and modeled after actual fly wheels. Some had solid 
rims, some jointed rims and some steel spokes. 

To give to the wheels the speed necessary for de- 
struction, use w^as made of a Dow steam ttu-bine capa- 
ble of being run at any speed up to loooo revolutions 
per minute. The turbine shaft was connected to the 
shaft carrying the fly wheels by a brass sleeve coup- 
ling loosely pinned to the shafts at each end in such a 
way as to form a universal joint, and so proportioned 
as to break or slip without injuring the turbine in case 
of sudden stoppage of the fly wheel shaft. 

One experiment with a shield made of two-inch 
plank convinced us that safety did not lie in that di- 
rection, and in succeeding experiments with the fifteen 
inch wheels a bomb-proof constructed of 6x12 inch 
white oak was used. The first experiment with a 
twenty-four inch wheel showed even this to be a flimsy 
contrivance. In subsequent experiments a shield made 
of 12x12 inch oak was used. Even this shield was split 
repeatedly and had to be re-enforced by bolts. 

A cast steel ring about four inches thick Uned with 
wooden blocks and covered wiih three inch oak plank- 
ing was finally adopted. 

The wheels were usually demolished by the ex- 
plosion. No crashing or rending noise was heard, only 
one quick, sharp report, like a musket shot. 

The following tables give a summary of a number 
of the experiments. 



lo6 



MACHINE DESIGN, 



TABLE XVilL — FIFTEEN INCH WHEELS. 





Bursting Speed. 


Centrifugal 










Tension 




No. 


Revs. 


Feet per 


v2 
10 


Remarks. 




per Minute. 


Second =v. 




I 


6,525 


430 


18,500 


Six arms. 


2 


6,525 


430 


18,500 


Six arms. 


3 


6,035 


395 


15,600 


Thin rim. 


4 


5,872 


380 


14,400 


Thin rim. 


5 


2,925 


192 


3,700 


Joint in rim. 


6 


5,600* 


368 


13,600 


Three arms. 


7 


6,198 


406 


16,500 


Three arms. 


8 


5,709 


368 


13,600 


Three arms. 


9 


5,709 


365 


13,300 


Thin rim. 


lO 


5.709 


361 


13,000 


Thin rim. 



* Doubtful. 



TABLE XIX.— TWENTY- FOUR INCH WHEELS. 





Shape and Size of Rim. 


Weight 


^0. 












of^ 




Diam- 


Breadth 


Depth 


Area 




Wheel, 




eter 






Sq. 


Style of Joint. 






Inches 


Inches 


Inches 


Inches 




Pounds. 


II 


24 


2H 


1-5 


3.18 


Solid rim. 


75.25 


12 


24 


4A 


.75 


3.85 


Internal flanges, bolted 


93. 


13 


24 


4 


.75 


3.85 


(( (( (( 


91-75 


14 


24 


4 


.75 


3.85 




95. 


15 


24 


4A 


.75 


3.85 




94.75 


16 


24 ^ 


1.2 


2.1 


2.45 


Three lugs and links 


65.1 


17 


24 


1.2 


2.1 


2.45 


Two lugs and links. 


65. 



MACHINE DESIGN. 

TABLE XX — FLANGES AND BOLTS. 



107 





Flanges. 


Bolts. 


Ko. 


Thick- 
ness, 

Inches. 


Effective 
Breadth, 

Inches. 


Effective 
Area, 

Inches. 


No. to 
each 
Joint. 


Diameter 
Inches. 


Total 

Tensile 

Strength, 

Pounds. 


12 

13 
14 
15 


if 


2.8 
2.75 
2.75 
2.5 


1.92 
2.34 


4 
4 
4 
4 




16,000 
16,000 
16,000 
20,000 



BY TESTING MACHINE. 



Tensile strength of cast iron = 19,600 pounds per square in. 
Transverse strength of cast iron = 46,600 pounds per square in. 
Tensile strength of -^^ bolts = 4,000 pounds. 
Tensile strength of | bolts = 5,000 pounds. 



TABLE XXI.— FAILURE OF FLANGED JOINTS. 





Area of Rim, 

Square Inches. 


Effect. Area 
flanges. Sq. Ins. 


Total Strength 
Bolts, Pounds. 


Bursting 
Speed. 


Cent. 
Tension. 




No. 


Rev. 

per 
Min. 


Ft.per 
Sec. 

= V. 


Per 
Sq.In. 

10 


Total 
Lbs. 


Remarks, 


II 


3.18 
3.85 

3.85 
3.85 






3,672 


385 


lA 800 


/It' rtnr\ 


Solid riin. 


12 


1.92 

,2:58 
2.34 


16,000 
16,000 
16,000 
20,000 




Flange broke. 
Flange broke. 
Bolts broke. 
Flange broke. 


13 

14 
15 


1,760 

1,875 
1,810 


184 
196 
190 


3,40013,100 
3,85014,800 
3,61013,900 



io8 



MACHINE DESIGN. 



TABLE XXII.— LINKED JOINTS. 







Lugs. 






Links. 




No. 






Area 
Sq. In. 




Effect 

Breadth. 
Inches. 


d CO 


Effective 
Area, 
Sq. Ids. 


i6 


.45 


I.O 


.45 


3 


•57 


.327 


.186 


17 


.44 


.98 


•43 


2 


.54 


.380 


.205 



Rim. 



Max. 

Area, 

Sq. Ins. 



2.45 
2.45 



Net 

Area, 

Sq. Ins. 



1.98 
1.98 



BY TESTING MACHINE. 

Tensile strength of cast iron = 19,600. 
Transverse strength of cast iron = 40,400. 
Av. tensile strength of each link = 10, 180. 



TABLE XXIII, — FAILURE OF LINKED JOINTS. 









Bursting 


Cent. 






i 


i 


Speed. 


Tension. 


















3 






Ft. 


Per 






No. 




0^ 


Rev. 
per 


per 
Sec. 


Sq.In. 
v2 


Total. 


Remarks. 




% 


^ 

-*-» 


Min. 


= v 


10 








tn 


t/5 












16 


30,540 


38,800 


3,060 


320 


10,240 


25,100 


Rim broke. 


17 


20,360 


38,800 


2,750 


290 


8,410 


20,600 


Lugs and Rim 
broke. 



MACHINE DESIGN. I09 

The flanged joirrts mentioned had the internal 
flanges and bolts common in large belt wheel rims 
while the linked joints were such as are commonly 
used in fly wheels not used for belts. 

Subsequent experiments have given approximately 
the same results as those just detailed. The highest 
velocity yet attained has been 424 feet per second; 
this is in a solid cast iron rim with numerous steel 
spokes. The average bursting velocity for solid cast 
rims with cast spokes is 400 feet per second. 

Wheels with jointed rims burst at speeds varying 
from 190 to 250 feet per second, according to the style 
of joint and its location. The following general con- 
clusions seem justified by these tests. 

1. Fly-wheels with solid rims, of the proportions 
usual among engine builders and having the usual 
number of arms, have a sufficient factor of safety at a 
rim speed of 100 feet per second if the iron is of good 
quality and there are no serious cooling strains. 

In such wheels the bending du*e to centrifugal 
force is slight, and may safely be disregarded. 

2. Rimjoints midway between the arms are a 
serious defect and reduce the factor of safety very ma- 
terially. Such joints are as serious mistakes in design 
as would be a joint in the middle of a girder under a 
heavy load. 

3. Joints made in the ordinary manner, with in- 
ternal flanges and bolts, are probably the worst that 
could be devised for this purpose. Under the most 
favorable circumstances they have only about one- 
fourth the strength of the solid rim and are particularly 
weak against bending. 

In several joints of this character, on large fly- 
wheels, calculation has shown a strength less than 
one-fifth that of the rim. 

4. The type of joint known as the link or prisoner 
joint is probably the best that could be devised for 



no MACHINE DESIGN. 

narrow rimmed wheels not intended to carry belts, 
and possesses when properly designed a strength about 
two-thirds that of the solid rim. 



66. Rims of Cast Iron Gears. A toothed wheel 
will burst at a less speed than a pulley because the 
teeth increase the weight and therefore the centrifugal 
force without adding to the strength. 

The centrifugal force and therefore the stresses 
due to the force will be increased nearly in the ratio 
that the weight of rim and teeth is greater than the 
weight of rim alone. 

This ratio in ordinary gearing varies from 1.5 to 
1.7. We will assume 1.6 as an average value. Neglect- 
ing bending we now have from equation (86) 

I2WV^ I9.2WV^ 

S=i.6x— ^= (90) 

and ^=S^ 

\19.2w 

= 326.2 ft. per second (91). 

Including bending 

«'=^-^^\^^+7^] (9^> 

As the transverse strength of cast iron by experi- 
ment is about double the tensile strength, a larger 
value of S may be allowed in formulas (88), (89) and 

(92). 

In built up wheels it is better to have tlie joints 
come near the arms to prevent the tendency of the 
bending to open the joints, and the fastenings should 
have the same tensile strength as the rim of the 
wheel. 



MACHINE DESIGN. Ill 

exampi.es. 

1 . Design eight arms of elliptic section for a gear 
48 ins. pitch diameter to transmit a pressure on tooth 
of 800 lbs. 

2. Determine bursting speed of the gear in pre- 
vious example in revolutions per minute if the thick- 
ness of rim is .75 inch. 

(i) Considering centrifugal tension alone. 

(2) Including bending of rim due to centrifugal 
force, assuming that one-half the stress due to bending 
is relieved by the stretching of the arms. 

3. Design a link joint for the rim of a fly-wheel, 
the rim being 8 ins. wide, 12 ins. deep and 18 ft. mean 
diameter, the links to have a tensile strength of 65000 
lbs. per sq. in. Determine the relative strength of joint 
and the probable bursting speed. 

4. Discuss the proportions of one of the following 
wheels in the laboratory and criticise dimensions. 

(a) Fly-wheel, AUis engine. 

(b) Fly - wheel, Fairbanks gas engine. 

(c) I^arge Medart pulleys, Electrical laboratory. 

(d) Belt - wheel, AUis engine. 



TRANSMISSION BY BELTS AND ROPES. 

67. Friction of Belting. The transmitting power 
of a belt is due to its friction on the pulley, and this 
friction is equal to the difference between the tensions 
of the driving and slack sides of the belt. 

Let w= width of belt. 

Ti=tension of driving 

side. 
T2= tension of slack side. 
R= friction of belt 

f = co-efficient of friction " ^> 

between belt and 

pulley. 
= arc of contact in cir- I 

cular measure. T 

Fig. 40. 

The tension T at any part of the arc of contact is 
intermediate between T^ and Tg. 

Let AB Fig. 40, be an indefinitely short element 
of the arc of contact, so that the tensions at A and B 
differ only by the amount dT. 

dT will then equal the friction on AB which we 
may call dR. 

Draw the intersecting tangents OT and OT' to 
represent the tensions and find their radial resultant 
OP. Then will OP represent the normal pressure on 
the arc AB which we will call P. 

<OTP= <ACB = d^ 
.-. P=Td^ 




B 



MACHINE DESIGN. 

The friction on AB is 

fP=:fTd^ 

or dT=dR=fTd<? 

dT 
and fd^ ~"T^ 

Integrating for the whole arc : 

/.T^dT T, 

T, id 
T"=e 

T,=^,=T,e- 
R=T,-T,=T,(i-e-f^) 



"3 



ie 



(93) 

The average value of f for leather belts on iron 
pulleys as determined by experiment is f=0.27 

If we denote the expression (i — e — ^^) by C, then 
for different arcs of contact C has the following values: 



Arc of 
Contact. 


go^' 


IIO° 


130° 


150'' 


180° 


210^ 


240° 


C 


•345 


.404 


•458 


.506 


•571 


.627 


.676 



The friction or force transmitted by a belt per 
inch of width is then 

R=CT, (94) 

and T^ must not exceed the safe working tensile 
strength of the material. 

A handy rule for calculating belts assumes C=.5 
which means that the force which a belt will transmit 
under ordinary conditions is one -half its tensile 
strength. 



114 MACHINE DESIGN . 

68 Strength of Belting. The strength of belting 
varies widely and only average values can be given. 
According to experiments made by the author good oak 
tanned belting has a breaking strength per inch of 
width as follows: 

Single. Double. 

Solid leather 900 lbs. 1400 lbs. 

Where riveted 600 lbs. 1200 lbs. 

Where laced 350 lbs. 

Canvas belting has approximately the same 
strength as leather. Tests of rubber coated canvas 
belts 4-ply, 8 inches wide, show a tensile strength of 
from 840 lbs to 930 lbs. per inch of width. 

69. Taylor's Experiments. The experiments of 
Mr. F. W. Taylor, as reported by him in Trans. Am. 
Soc. Mech. Eng. Vol. XV. afFord the most valuable 
data now available on the performance of belts in actual 
service. 

These experiments were carried on during a period 
of nine years at the Midvale Steel Works. Mr. Tay- 
lor's conclusions may be epitomized as follows : 

1 . Narrow double belts are more economical than 
single ones of a greater width. 

2. All joints should be spliced and cemented. 

3. The most economical belt speed is from 4000 to 
4500 ft. per min. 

4. The working tension of a double belt should 
not exceed 35 lbs. per inch of width, but the belt may 
be first tightened to about double this. 

5. Belts should be cleaned and greased every six 
months. 

6. The best length is from 20 to 25 feet between 
centers. 



MACHINE DKSIGN. II5 

70. Rules for Width of Belts It will be noticed 
that Mr. Taylor recommends a working tension only to 
to A- the breaking strength of the belt. He justifies 
this by saying that belts so designed gave much less 
trouble from stoppage and repairs and were conse- 
quently more economical than those designed by the 
ordinary rules. 

In the following formulas 50 lbs. per inch of width 
is allowed for double belts and 30 lbs. for single belts. 
These are suitable values for belts which are not run- 
ning continuously. The formulas may be easily changed 
for other thicknesses and for other values of CT^. 

Let HP=horse power transmitted. 

D=diameter of driving pulley in inches. 
N=no. revs, per min. of pulley. 

The moment of force transmitted by belt is 

RD _ CT.wD _ 
2 2 ~ 

TN CT.wDN 
^^^ ^^=6^S^ 126050 (95) 

Substituting the values assumed for CT, and solving 

for w : 

HP 
Single belts w = 4200 ^-^ (96) 

HP 
Doublebelts w= 2500 ^^ (97) 

The most convenient rules for belting are those 
which give the horse-power of a belt in terms of the 
surface passing a fixed point per minute. 

In formula (95) CT.wDN 

^^== 126050 

we will substitute the following : 



Il6 MACHINE DESIGN. 

w 
W= width of belt in feet = — 

12 

V= velocity in ft. per mm. = 



or HP: 



i44CT,WV 



1260507: 

Substituting values of C and 1\ as before and 
solving for WV=square feet per minute we have ap- 
proximately : 

Single belts WV= 90 HP (98) 

Double belts WV= 55 HP (99) 



71. Speed of Belting. As in the case of pulley 
rims, so in that of belts a certain amount of tension is 
caused by the centrifugal force of the belt as it passes 
around the pulley. 

From equation (84) i2wv^ 

where v= velocity in ft. per sec. 

\v= weight of material per cu. in. 
S = tensile stress per sq. in. 

To make this formula more convenient for use we 
will make the following changes in the constants: 

Let V = velocity of belt in ft. per minute = 6ov. 
w= weight of ordinary belting. 

= .032 per cu. in. 
Si= tensile stress per inch width, caused by 

centrifugal force. 
=about TE S for single belts. 

V 

Then v=^ 

16S, 



MACHINE DESIGN. 1 1 7 

Substituting these values in (86) and solving for S^ 

Si= I 610000 (^o^) 

The speed usually given as a safe limit for ordi- 
nary belts is 3000 ft. permin., but belts are sometimes 
run at a speed exceeding 6000 ft. per min. 

Substituting different values of V in the formula 
we have : 

V=30oo Si= 5.59 lbs. 

V=4ooo Si= 9.94 lbs. 

V=5ooo Si = 15 -53 lbs. 

V=6ooo 81=22.36 lbs. 

The values of Si for double belts will be nearly 
twice those given above. At a speed of 5000 ft. per 
minute the maximum tension per inch of width on a 
single belt designed by formula ( 96 ) , if we call 
C = .5, will be : 

(3oX2) + i5.= 75 lbs. 

giving a factor of safety of eight or ten at the splices. 

In a similar manner we find the maximum tension 
per inch of width of a double belt to be : 

(50x2) +30= 130 lbs. 

and the margin of safety about the same as in single 
belting. 

72. Manila Rope Transmission. Ropes are some- 
times used instead of fiat belts for transmitting power 
short distances. They possess the following advan- 
tages: they are cheaper than belts in first cost ; chey 
are flexible in every direction and can De carried 
around corners readily. They have however the dis- 
advantage of being less efiicient in transmission than 
leather belts and less durable; they are also somewhat 
diflEicult to splice or repair. 



Il8 MACHINE DESIGN. 

There are two systems of rope driving in common 
use : the English and the American. In the former 
there are as many separate ropes as there are grooves 
in one pulley, each rope being an endless loop always 
running in one groove. 

In the American system one continuous rope is 
used passing back and forth from one groove to an- 
other and finally returning to the starting point. 

The advantage of the English system consists in 
the fact that one of the ropes may fail without causing 
a breakdown of the entire drive, there usually being 
two or three ropes in excess of the number actually 
necessary. On the other hand the American system 
has the advantage of a uniform regulation of the ten- 
sion on all the plies of rope. The guide pulley, which 
guides the last slack turn of rope back to the starting 
point, is usually also a tension pulley and can be 
weighted to secure any desired tension. The English 
method is most used for heavy drives from engines to 
head shafts ; the American for lighter work in dis- 
tributing power to the different rooms of a factory. 
The grooves in the pulle^^s for manila or cotton ropes 
usually have their sides inclined at an angle of about 
45°, thus wedging the rope in the groove. 

The Walker groove has curved sides as shown 
in Fig. 41, the curvature increasing towards the 
bottom. As the rope 
wears and stretches it 
becomes smaller and 
sinks deeper in the 
groove; the sides of the 
groove being more ob- 
lique near the bottom, 
the older rope is not 
pinched so hard as the 
newer and this tends 
to throw more of the 
work on the latter. Fig. 41 




MACHINE DESIGN. II 9 

73. Strength of Manila Ropes. The formulas for 
transmission by ropes are similar to those for belts the 
values for S and ^ being changed. The ultimate tensile 
strength of manila and hemp rope is about loooo lbs. 
per sq. in. 

To insure durability and efficiency it has been 
found best in practice to use a large factor of safety. 
Prof. Forrest R. Jones in his book on Machine Design 
recommends a maximum tension of 200 d^ pounds 
where d is the diameter of rope in inches. This cor- 
responds to a tensile stress of 255 lbs. per sq. in. or a 
factor of safety of about 40. 

The value of f for manila on metal is about 0.12, 
but as the normal pressure between the two surfaces 
is increased by the wedge action of the rope in the 
groove we shall have the apparent value of f : 

a 
r = f-r-sin"^ where 

a = angle of groove, 
For a = 45° to 30'' 

f^ varies from 0,3 to 0.5 and these values should be 
used in formula (93). 

(i — e"~f^) in this formula for an arc of contact of 
150°, becomes either .54 or .73 according as T is taken 
0.3 or 0.5. 

If Tj is assumed as 250 lbs. per sq. in., the force 
R transmitted b^^ the rope varies from 135 lbs to 185 
lbs. per sq. in. area of rope section. 

The following table gives the horse -power of 
manila ropes based on a maximum tension of 255 
lbs. per sq. in. 



I20 



MACHINE DESIGN. 



TABLE XXIV. 

Table of the horse-power of transmission rope, reprinted 
from the transactions of the American Society of Mechanical 
Engineers, Vol. 12, page 230, Article on *'Rope Driving" by 
C. W. Hunt. 

The working strain is 800 lbs. for a 2 inch diameter rope 
and is the same at all speeds, due allowance having been made 
for loss by centrifugal force. 



n i 


SPEED OF THE ROPE IN FEET PER MINUTE. 


1 




1500 


2000 


2500 


3000 


3500 


4000 


4500 


5000 


6000 


7000 


Smalle 
Diam. 
leys, I 


fittl 


3-3 


4.3 


5.2 


5.8 


6.7 


7.2 


7.7 


7.7 


7-1 




^ 


4.9 


30 


H 


4.5 


5.9 


7.0 


8.2 


9.1 


9.8 


10.8 


10.8 


9-3 


6.9 


36 


I. 


5.8 


7.7 


9.2 


10.7 


11.9 


12.8 


13.6 


13-7 


12.5 


8.8 


42 


iX 


9.2 


12. 1 


14.3 


16.8 


18.6 


20.0 


21.2 


21.4 


19.5 


13.8 


54 


I'A 


I3-I 


17.4 


20.7 


23.1 


26.8 


28.8 


30.6 


30.8 


28.2 


19.8 


60 


iX 


18.0 


23.7 


28.2 


32.8 


36.4 


39-2 


41.5 


41.8 


37.4 


27.6 


72 


2 


23.1 


30.8 


36.8 


42.8 


47.6 


51.2 


54.4 


54.8 


50.0 


35-2 


84 



74. Wire Rope Transmission. Wire Ropes have 
been used to transmit power where the distances were 
too great for belting or hemp rope transmission. The 
increased use of electrical transmission is gradually 
crowding out this latter form of rope driving. 

For comparatively short distances of from 100 to 
500 yards wire rope still offers a cheap and simple 
means of carrying power. 

The pulleys or wheels are entirely different from 
those used with manila ropes. 



MACHINE DESIGN, 



121 



Fig. 42 shows a section of the rim of such a pul- 
ley. The rope does not touch the sides of the groove 
but rests on a shallow depression in a wooden, leather 
or rubber filling at the bottom. The 
high side flanges prevent the rope 
from leaving the pulley when sway- 
ing on account of the high speed. 



The pulleys must be large, usu- 
ally about 100 times the diameter of 
rope used, and run at comparatively 
high speeds. The ropes should not 
be less than 200 feet long unless 
some form of tightening pulley is 
used. — Table XXV is taken from 
Roebling. 




Fig. 42. 



lyong ropes should be supported by idle pulleys 
every 400 feet. 

EXAMPLES. 



1. Design a main driving belt to transmit 150 HP 
from a belt wheel 18 ft. in diameter and making 80 
revs, per min. The belt to be double leather without 
rivets. 

2. Investigate driving belt on Allis engine and 
calculate the horse - power it is capable of transmit- 
ting economically. 

3. Calculate the total maximum tension per inch 
of width due to load and to centrifugal force of the 
driving belt on the generator used for lighting, assum- 
ing the maximum load to be 50 HP. 

4. Design a manila rope drive, English system, to 
transmit 500 HP, the wheel on the engine being 20 
feet in diameter and making 60 revs, per min. Use 



122 



MACHINE DESIGN. 



TABLE XXV. 
TRANSMISSION OF POWER BY WIRE ROPES. 

Showing necessary size and speed of wheels and rope to 
obtain any desired amount of power. 



Diameter of 
Wheel in ft. 


Number of 
Revolutions. 


Diameter 
of Rope. 


Horse - 
Power. 


«4-l 

on. 

0^ 


Number of 
Revolutions. 


Diameter 
of Rope. 


Horse - 
Power. 


4 


8o 


3/8 


3-3 


10 


80 


II-16 


58.4 




lOO 


3/8 


4.1 




100 


II-16 


73- 




I20 


3/8 


5- 




120 


II-16 


87.6 




140 


3/8 


5-8 




140 


II-16 


102.2 


5 


80 


7-16 


6.9 


II 


80 


II-16 


75-5 




100 


7-16 


8.6 




100 


II-16 


94-4 




120 


7-16 


10.3 




120 


II-16 


II3-3 




140 


7-16 


12. 1 




140 


II-16 


132. 1 


6 


80 


V2 


10.7 


12 


80 


V\ 


99-3 




100 


Vz 


13-4 




100 


V\ 


1 24. 1 




i.?o 


% 


16.1 




120 


% 


148.9 




140 


V2 


18.7 




140 


Ya 


173-7 


7 


80 


9-16 


16.9 


13 


80 


Vx 


122.6 




100 


9-t6 


21. 1 




100 


Ya 


153-2 




120 


9-16 


25-3 




120 


Vx 


183-9 


8 


80 


/8 


22. 


14 


80 


Vz 


148. 




ICX> 


^8 


27-5 




100 


/s 


185. 




120 


5/8 


33 -o 




120 


/8 


222. 


9 


80 


Vz 


41-5 


15 


80 


^ 


217. 




100 


Vz 


51-9 




100 


Vz 


259- 




120 


Vz 


62.2 




120 


n 


300- 



MACHINE DESIGN. 1 23 

Hunt's table and then check by calculating the centri- 
fugal tension and the total maximum tension on each 
rope. 

5. Design a wire rope transmission to carry 
1 20 H P a distance of one - quarter mile using two 
ropes. Determine working and maximum tension 
on rope, length of rope, diameter and speed of pulleys 
and number of supporting pulleys. 



INDEX. 



PAGE. 

Adjustment of Bearings 61-62 

Ball Bearings 76-80 

Barlow's Formula 17 

Beams, Bending 7-8 

Deflection 11 

Bearings, Adjustment 61-62 

Ball 76-80 

Ivubrication 63-64 

Roller 80-82 

Rotating , 61-67 

Sliding 53-57 

Thrust 74 

Belting, Friction of 112-113 

Experiments 114 

Speeds 116 

Strength 114 

Width 115-116 

Boiler Shells 16 

Bolts and Nuts 39 

Coupling 86 

Strength, Table 40 

Bursting Fly-Wheels 105-109 

Caps and Bolts 68 

Columns, Strength 8-9 

Cotters 50-51 

Couplings, Shaft 85-86 

Cylinders, Steam 20-24 

Table 22 

Deflection, Formulas 11 

Design, Principles of . 12-13 



Fly- Wheels, Experiments 105-109 

Formulas, General 7~9 

Frame Design 12-13 

Frames, Machine 28-30 

Friction of Belts 112-113 

Journals 65 

Pivots 70-73 

Gearing 100 

Arms and Rim loo-ioi 

Bevel 98 

Safe Speed no 

Spur 91-98 

Guides 57 

Hangers 88 

Heating of Journals 66 

Hooks 41 

Hyatt Rollers 81 

Iron, Cast 4 

Malleable 4 

Wrought 3 

Joints, Riveted 42-48 

Butt 45-46 

I/ap 43-44 

Tables 47-48 

Joint Pins 49 

Journals, Friction . 65 

Heating 66 

Pressure 65 

Strength 67 

Keys, Cotter 50-51 

Shafting 87 

Ivamc's Formulas 19 

Lubrication 63-64 

Materials of Construction 3 

Notation Used 7 

Nuts, Check 41 

Pipe, Table 18 

Pivots, Conical 71 

Flat • 70 

Schiele 72-73 



Plates, Flat 24-27 

Experiments 26-27 

Pulleys, Arms of loo-ioi 

Safe Speed 102-104 

Riveted Joints 42-49 

Roller Bearings 80-82 

Rope, Manila 117-120 

Horse - Power 120 

Strength 119 

Rope, Wire 120-122 

Tables 122 

Sections, Cored 28-29 

Section Moduli 10 

Shells, Thin 16 

Thick 17-20 

Shafting 83-85 

Slides, General 53 

Angular 54 

Flat 56 

Gibbed 55 

Speed, Safe 102-104 

Springs, Experiments 33-34 

I^lat 36-37 

Tension and Compression 31-34 

Torsion 34-35 

Steel 3-4 

Stress and Strain 2; 

Strength of Metals 5-^' 

Stuffing Boxes 57-58 

Experiments 59-^' 

Supports, Machine 14 

Teeth of Gears 91-99 

Experiments 96-9S 

Thrust Bearings 74 

Units Used •• 3 



TEXT BOOKS 

BY 

Charles H. Benjamin, M. E., 

» 

Professor of Mechanical Engineering, 
CASE SCHOOL OF APPLIED SCIENCE. 



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